Matrix (mathematics)
Encyclopedia
In mathematics
, a matrix (plural matrices, or less commonly matrixes) is a rectangular array of numbers, symbols, or expressions. The individual items in a matrix are called its elements or entries. An example of a matrix with six elements is
Matrices of the same size can be added
or subtracted element by element. The rule for matrix multiplication
is more complicated, and two matrices can be multiplied only when the number of columns in the first equals the number of rows in the second. A major application of matrices is to represent linear transformation
s, that is, generalizations of linear functions such as . For example, the rotation
of vectors in three dimensional space is a linear transformation. If R is a rotation matrix and v is a column vector (a matrix with only one column) describing the position of a point in space, the product Rv is a column vector describing the position of that point after a rotation. The product of two matrices is a matrix that represents the composition
of two linear transformations. Another application of matrices is in the solution of a system of linear equations. If the matrix is square, it is possible to deduce some of its properties by computing its determinant
. For example, a square matrix has an inverse if and only if its determinant is not zero. Eigenvalues and eigenvectors provide insight into the geometry of linear transformations.
Matrices find applications in most scientific fields. In physics
, matrices are used to study electrical circuits, optics, and quantum mechanics
. In computer graphics
, matrices are used to project a 3dimensional image onto a 2dimensional screen, and to create realisticseeming motion. Matrix calculus
generalizes classical analytical
notions such as derivative
s and exponentials to higher dimensions.
A major branch of numerical analysis
is devoted to the development of efficient algorithms for matrix computations, a subject that is centuries old and is today an expanding area of research. Matrix decomposition methods simplify computations, both theoretically and practically. Algorithms that are tailored to the structure of particular matrix structures, e.g. sparse matrices
and neardiagonal matrices
, expedite computations in finite element method
and other computations. Infinite matrices occur in planetary theory and in atomic theory. A simple example is the matrix representing the derivative operator, which acts on the Taylor series
of a function.
arrangement of mathematical expressions that can be simply number
s. For example,
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, a matrix (plural matrices, or less commonly matrixes) is a rectangular array of numbers, symbols, or expressions. The individual items in a matrix are called its elements or entries. An example of a matrix with six elements is
Matrices of the same size can be added
Matrix addition
In mathematics, matrix addition is the operation of adding two matrices by adding the corresponding entries together. However, there are other operations which could also be considered as a kind of addition for matrices, the direct sum and the Kronecker sum....
or subtracted element by element. The rule for matrix multiplication
Matrix multiplication
In mathematics, matrix multiplication is a binary operation that takes a pair of matrices, and produces another matrix. If A is an nbym matrix and B is an mbyp matrix, the result AB of their multiplication is an nbyp matrix defined only if the number of columns m of the left matrix A is the...
is more complicated, and two matrices can be multiplied only when the number of columns in the first equals the number of rows in the second. A major application of matrices is to represent linear transformation
Linear transformation
In mathematics, a linear map, linear mapping, linear transformation, or linear operator is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication. As a result, it always maps straight lines to straight lines or 0...
s, that is, generalizations of linear functions such as . For example, the rotation
Rotation (mathematics)
In geometry and linear algebra, a rotation is a transformation in a plane or in space that describes the motion of a rigid body around a fixed point. A rotation is different from a translation, which has no fixed points, and from a reflection, which "flips" the bodies it is transforming...
of vectors in three dimensional space is a linear transformation. If R is a rotation matrix and v is a column vector (a matrix with only one column) describing the position of a point in space, the product Rv is a column vector describing the position of that point after a rotation. The product of two matrices is a matrix that represents the composition
Function composition
In mathematics, function composition is the application of one function to the results of another. For instance, the functions and can be composed by computing the output of g when it has an argument of f instead of x...
of two linear transformations. Another application of matrices is in the solution of a system of linear equations. If the matrix is square, it is possible to deduce some of its properties by computing its determinant
Determinant
In linear algebra, the determinant is a value associated with a square matrix. It can be computed from the entries of the matrix by a specific arithmetic expression, while other ways to determine its value exist as well...
. For example, a square matrix has an inverse if and only if its determinant is not zero. Eigenvalues and eigenvectors provide insight into the geometry of linear transformations.
Matrices find applications in most scientific fields. In physics
Physics
Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...
, matrices are used to study electrical circuits, optics, and quantum mechanics
Quantum mechanics
Quantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particlelike and wavelike behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...
. In computer graphics
Computer graphics
Computer graphics are graphics created using computers and, more generally, the representation and manipulation of image data by a computer with help from specialized software and hardware....
, matrices are used to project a 3dimensional image onto a 2dimensional screen, and to create realisticseeming motion. Matrix calculus
Matrix calculus
In mathematics, matrix calculus is a specialized notation for doing multivariable calculus, especially over spaces of matrices, where it defines the matrix derivative. This notation was to describe systems of differential equations, and taking derivatives of matrixvalued functions with respect...
generalizes classical analytical
Mathematical analysis
Mathematical analysis, which mathematicians refer to simply as analysis, has its beginnings in the rigorous formulation of infinitesimal calculus. It is a branch of pure mathematics that includes the theories of differentiation, integration and measure, limits, infinite series, and analytic functions...
notions such as derivative
Derivative
In calculus, a branch of mathematics, the derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a...
s and exponentials to higher dimensions.
A major branch of numerical analysis
Numerical analysis
Numerical analysis is the study of algorithms that use numerical approximation for the problems of mathematical analysis ....
is devoted to the development of efficient algorithms for matrix computations, a subject that is centuries old and is today an expanding area of research. Matrix decomposition methods simplify computations, both theoretically and practically. Algorithms that are tailored to the structure of particular matrix structures, e.g. sparse matrices
Sparse matrix
In the subfield of numerical analysis, a sparse matrix is a matrix populated primarily with zeros . The term itself was coined by Harry M. Markowitz....
and neardiagonal matrices
Diagonal matrix
In linear algebra, a diagonal matrix is a matrix in which the entries outside the main diagonal are all zero. The diagonal entries themselves may or may not be zero...
, expedite computations in finite element method
Finite element method
The finite element method is a numerical technique for finding approximate solutions of partial differential equations as well as integral equations...
and other computations. Infinite matrices occur in planetary theory and in atomic theory. A simple example is the matrix representing the derivative operator, which acts on the Taylor series
Taylor series
In mathematics, a Taylor series is a representation of a function as an infinite sum of terms that are calculated from the values of the function's derivatives at a single point....
of a function.
Definition
A matrix is a rectangularRectangle
In Euclidean plane geometry, a rectangle is any quadrilateral with four right angles. The term "oblong" is occasionally used to refer to a nonsquare rectangle...
arrangement of mathematical expressions that can be simply number
Number
A number is a mathematical object used to count and measure. In mathematics, the definition of number has been extended over the years to include such numbers as zero, negative numbers, rational numbers, irrational numbers, and complex numbers....
s. For example,

An alternative notation uses large parentheses instead of box brackets:

The horizontal and vertical lines in a matrix are called rows and columns, respectively. The numbers in the matrix are called its entries or its elements. To specify the size of a matrix, a matrix with m rows and n columns is called an mbyn matrix or m × n matrix, while m and n are called its dimensions. The above is a 4by3 matrix.
A matrix with one row (a 1 × n matrix) is called a row vector, and a matrix with one column (an m × 1 matrix) is called a column vector. Any row or column of a matrix determines a row or column vector, obtained by removing all other rows or columns respectively from the matrix. For example, the row vector for the third row of the above matrix A is
When a row or column of a matrix is interpreted as a value, this refers to the corresponding row or column vector. For instance one may say that two different rows of a matrix are equal, meaning they determine the same row vector. In some cases the value of a row or column should be interpreted just as a sequence of values (an element of R^{n} if entries are real numbers) rather than as a matrix, for instance when saying that the rows of a matrix are equal to the corresponding columns of its transposeTransposeIn linear algebra, the transpose of a matrix A is another matrix AT created by any one of the following equivalent actions:...
matrix.
Most of this article focuses on real and complex matrices, i.e., matrices whose elements are real or complex, respectively. More general types of entries are discussed below.
Notation
The specifics of matrices notation varies widely, with some prevailing trends. Matrices are usually denoted using uppercase letters, while the corresponding lowercase letters, with two subscript indices, represent the entries. In addition to using uppercase letters to symbolize matrices, many authors use a special typographical styleEmphasis (typography)In typography, emphasis is the exaggeration of words in a text with a font in a different style from the rest of the text—to emphasize them. Methods and use :...
, commonly boldface upright (nonitalic), to further distinguish matrices from other mathematical objects. An alternative notation involves the use of a doubleunderline with the variable name, with or without boldface style, (e.g., ).
The entry in the ith row and the jth column of a matrix is typically referred to as the i,j, (i,j), or (i,j)^{th} entry of the matrix. For example, the (2,3) entry of the above matrix A is 7. The (i, j)^{th} entry of a matrix A is most commonly written as a_{i,j}. Alternative notations for that entry are A[i,j] or A_{i,j}.
Sometimes a matrix is referred to by giving a formula for its (i,j)^{th} entry, often with double parenthesis around the formula for the entry, for example, if the (i,j)^{th} entry of A were given by a_{ij}, A would be denoted ((a_{ij})).
An asterisk is commonly used to refer to whole rows or columns in a matrix. For example, a_{i,∗} refers to the i^{th} row of A, and a_{∗,j} refers to the j^{th} column of A. The set of all mbyn matrices is denoted (m, n).
A common shorthand is A = [a_{i,j}]_{i = 1,...,m; j = 1,...,n} or more briefly A = [a_{i,j}]_{m×n}
to define an m × n matrix A. Usually the entries a_{i,j} are defined separately for all integers and . They can however sometimes be given by one formula; for example the 3by4 matrix
can alternatively be specified by A = [i − j]_{i = 1,2,3; j = 1,...,4}, or simply A = ((ij)), where the size of the matrix is understood.
Some programming languages start the numbering of rows and columns at zero, in which case the entries of an mbyn matrix are indexed by and . This article follows the more common convention in mathematical writing where enumeration starts from 1.
Basic operations
There are a number of operations that can be applied to modify matrices called matrix addition, scalar multiplication and transposition. These form the basic techniques to deal with matrices.Operation Definition Example Addition The sum A+B of two mbyn matrices A and B is calculated entrywise:_{i,j} = A_{i,j} + B_{i,j}, where 1 ≤ i ≤ m and 1 ≤ j ≤ n.
Scalar multiplication The scalar multiplication cA of a matrix A and a number c (also called a scalar Scalar (mathematics)In linear algebra, real numbers are called scalars and relate to vectors in a vector space through the operation of scalar multiplication, in which a vector can be multiplied by a number to produce another vector....
in the parlance of abstract algebraAbstract algebraAbstract algebra is the subject area of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras...
) is given by multiplying every entry of A by c: (cA)_{i,j} = c · A_{i,j}.
Transpose The transpose of an mbyn matrix A is the nbym matrix A^{T} (also denoted A^{tr} or ^{t}A) formed by turning rows into columns and vice versa:  (A^{T})_{i,j} = A_{j,i}.
Familiar properties of numbers extend to these operations of matrices: for example, addition is commutative, i.e., the matrix sum does not depend on the order of the summands: A + B = B + A.
The transpose is compatible with addition and scalar multiplication, as expressed by (cA)^{T} = c(A^{T}) and (A + B)^{T} = A^{T} + B^{T}. Finally, (A^{T})^{T} = A.
Row operations are ways to change matrices. There are three types of row operations: row switching, that is interchanging two rows of a matrix; row multiplication, multiplying all entries of a row by a nonzero constant; and finally row addition, which means adding a multiple of a row to another row. These row operations are used in a number of ways including solving linear equations and finding inverses.
Matrix multiplication, linear equations and linear transformations
Multiplication of two matrices is defined only if the number of columns of the left matrix is the same as the number of rows of the right matrix. If A is an mbyn matrix and B is an nbyp matrix, then their matrix product AB is the mbyp matrix whose entries are given by dot productDot productIn mathematics, the dot product or scalar product is an algebraic operation that takes two equallength sequences of numbers and returns a single number obtained by multiplying corresponding entries and then summing those products...
of the corresponding row of A and the corresponding column of B:
where 1 ≤ i ≤ m and 1 ≤ j ≤ p. For example, the underlined entry 1 in the product is calculated as
Matrix multiplication satisfies the rules (AB)C = A(BC) (associativityAssociativityIn mathematics, associativity is a property of some binary operations. It means that, within an expression containing two or more occurrences in a row of the same associative operator, the order in which the operations are performed does not matter as long as the sequence of the operands is not...
), and (A+B)C = AC+BC as well as C(A+B) = CA+CB (left and right distributivityDistributivityIn mathematics, and in particular in abstract algebra, distributivity is a property of binary operations that generalizes the distributive law from elementary algebra.For example:...
), whenever the size of the matrices is such that the various products are defined. The product AB may be defined without BA being defined, namely if A and B are mbyn and nbyk matrices, respectively, and Even if both products are defined, they need not be equal, i.e., generally one has AB ≠ BA,
i.e., matrix multiplication is not commutativeCommutativityIn mathematics an operation is commutative if changing the order of the operands does not change the end result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it...
, in marked contrast to (rational, real, or complex) numbers whose product is independent of the order of the factors. An example of two matrices not commuting with each other is:
whereas
The identity matrixIdentity matrixIn linear algebra, the identity matrix or unit matrix of size n is the n×n square matrix with ones on the main diagonal and zeros elsewhere. It is denoted by In, or simply by I if the size is immaterial or can be trivially determined by the context...
I_{n} of size n is the nbyn matrix in which all the elements on the main diagonal are equal to 1 and all other elements are equal to 0, e.g.
It is called identity matrix because multiplication with it leaves a matrix unchanged: MI_{n} = I_{m}M = M for any mbyn matrix M.
Besides the ordinary matrix multiplication just described, there exist other less frequently used operations on matrices that can be considered forms of multiplication, such as the Hadamard product and the Kronecker productKronecker productIn mathematics, the Kronecker product, denoted by ⊗, is an operation on two matrices of arbitrary size resulting in a block matrix. It gives the matrix of the tensor product with respect to a standard choice of basis. The Kronecker product should not be confused with the usual matrix...
. They arise in solving matrix equations such as the Sylvester equationSylvester equationThe Sylvester equation, commonly encountered in control theory, is the matrix equation of the formA X + X B = C,where A,B,X,C are n \times n matrices. A,B,C are known...
.
Linear equations
A particular case of matrix multiplication is tightly linked to linear equations: if x designates a column vector (i.e., n×1matrix) of n variables x_{1}, x_{2}, ..., x_{n}, and A is an mbyn matrix, then the matrix equation Ax = b,
where b is some m×1column vector, is equivalent to the system of linear equations A_{1,1}x_{1} + A_{1,2}x_{2} + ... + A_{1,n}x_{n} = b_{1}
 ...
 A_{m,1}x_{1} + A_{m,2}x_{2} + ... + A_{m,n}x_{n} = b_{m} .
This way, matrices can be used to compactly write and deal with multiple linear equations, i.e., systems of linear equations.
Linear transformations
Matrices and matrix multiplication reveal their essential features when related to linear transformations, also known as linear maps. A real mbyn matrix A gives rise to a linear transformation R^{n} → R^{m} mapping each vector x in R^{n} to the (matrix) product Ax, which is a vector in R^{m}. Conversely, each linear transformation f: R^{n} → R^{m} arises from a unique mbyn matrix A: explicitly, the of A is the i^{th} coordinate of f(e_{j}), where e_{j} = (0,...,0,1,0,...,0) is the unit vector with 1 in the j^{th} position and 0 elsewhere. The matrix A is said to represent the linear map f, and A is called the transformation matrix of f.
For example, the 2×2 matrix
can be viewed as the transform of the unit squareUnit squareIn mathematics, a unit square is a square whose sides have length 1. Often, "the" unit square refers specifically to the square in the Cartesian plane with corners at , , , and .In the real plane:...
into a parallelogramParallelogramIn Euclidean geometry, a parallelogram is a convex quadrilateral with two pairs of parallel sides. The opposite or facing sides of a parallelogram are of equal length and the opposite angles of a parallelogram are of equal measure...
with vertices at , , , and . The parallelogram pictured at the right is obtained by multiplying A with each of the column vectors and in turn. These vectors define the vertices of the unit square.
The following table shows a number of 2by2 matrices with the associated linear maps of R^{2}. The blue original is mapped to the green grid and shapes, the origin (0,0) is marked with a black point.Horizontal shear with m=1.25. Horizontal flip Squeeze mapping Squeeze mappingIn linear algebra, a squeeze mapping is a type of linear map that preserves Euclidean area of regions in the Cartesian plane, but is not a Euclidean motion.For a fixed positive real number r, the mapping →...
with r=3/2Scaling Scaling (geometry)In Euclidean geometry, uniform scaling is a linear transformation that enlarges or shrinks objects by a scale factor that is the same in all directions. The result of uniform scaling is similar to the original...
by a factor of 3/2Rotation by π/6^{R} = 30°
Under the 1to1 correspondenceBijectionA bijection is a function giving an exact pairing of the elements of two sets. A bijection from the set X to the set Y has an inverse function from Y to X. If X and Y are finite sets, then the existence of a bijection means they have the same number of elements...
between matrices and linear maps, matrix multiplication corresponds to compositionFunction compositionIn mathematics, function composition is the application of one function to the results of another. For instance, the functions and can be composed by computing the output of g when it has an argument of f instead of x...
of maps: if a kbym matrix B represents another linear map g : R^{m} → R^{k}, then the composition is represented by BA since(x) = g(f(x)) = g(Ax) = B(Ax) = (BA)x.
The last equality follows from the abovementioned associativity of matrix multiplication.
The rank of a matrix A is the maximum number of linearly independentLinear independenceIn linear algebra, a family of vectors is linearly independent if none of them can be written as a linear combination of finitely many other vectors in the collection. A family of vectors which is not linearly independent is called linearly dependent...
row vectors of the matrix, which is the same as the maximum number of linearly independent column vectors. Equivalently it is the dimension of the imageImage (mathematics)In mathematics, an image is the subset of a function's codomain which is the output of the function on a subset of its domain. Precisely, evaluating the function at each element of a subset X of the domain produces a set called the image of X under or through the function...
of the linear map represented by A. The ranknullity theoremRanknullity theoremIn mathematics, the rank–nullity theorem of linear algebra, in its simplest form, states that the rank and the nullity of a matrix add up to the number of columns of the matrix. Specifically, if A is an mbyn matrix over some field, thenThis applies to linear maps as well...
states that the dimension of the kernel of a matrix plus the rank equals the number of columns of the matrix.
Square matrices
A square matrix is a matrix with the same number of rows and columns. An nbyn matrix is known as a square matrix of order n. Any two square matrices of the same order can be added and multiplied. A square matrix A is called invertible or nonsingular if there exists a matrix B such that AB = I_{n}.
This is equivalent to BA = I_{n}. Moreover, if B exists, it is unique and is called the inverse matrix of A, denoted A^{−1}.
The entries A_{i,i} form the main diagonal of a matrix. The trace, tr(A) of a square matrix A is the sum of its diagonal entries. While, as mentioned above, matrix multiplication is not commutative, the trace of the product of two matrices is independent of the order of the factors: tr(AB) = tr(BA).
Also, the trace of a matrix is equal to that of its transpose, i.e., tr(A) = tr(A^{T}).
If all entries outside the main diagonal are zero, A is called a diagonal matrixDiagonal matrixIn linear algebra, a diagonal matrix is a matrix in which the entries outside the main diagonal are all zero. The diagonal entries themselves may or may not be zero...
. If only all entries above (below) the main diagonal are zero, A is called a lower triangular matrixTriangular matrixIn the mathematical discipline of linear algebra, a triangular matrix is a special kind of square matrix where either all the entries below or all the entries above the main diagonal are zero...
(upper triangular matrix, respectively). For example, if n = 3, they look like
(diagonal), (lower) and (upper triangular matrix).
Determinant
The determinant det(A) or A of a square matrix A is a number encoding certain properties of the matrix. A matrix is invertible if and only ifIf and only ifIn logic and related fields such as mathematics and philosophy, if and only if is a biconditional logical connective between statements....
its determinant is nonzero. Its absolute valueAbsolute valueIn mathematics, the absolute value a of a real number a is the numerical value of a without regard to its sign. So, for example, the absolute value of 3 is 3, and the absolute value of 3 is also 3...
equals the area (in R^{2}) or volume (in R^{3}) of the image of the unit square (or cube), while its sign corresponds to the orientation of the corresponding linear map: the determinant is positive if and only if the orientation is preserved.
The determinant of 2by2 matrices is given by
When the determinant is equal to one, then the matrix represents an equiareal mapping. The determinant of 3by3 matrices involves 6 terms (rule of SarrusRule of SarrusSarrus' rule or Sarrus' scheme is a method and a memorization scheme to compute the determinant of a 3×3 matrix. It is named after the French mathematician Pierre Frédéric Sarrus....
). The more lengthy Leibniz formula generalises these two formulae to all dimensions.
The determinant of a product of square matrices equals the product of their determinants: det(AB) = det(A) · det(B). Adding a multiple of any row to another row, or a multiple of any column to another column, does not change the determinant. Interchanging two rows or two columns affects the determinant by multiplying it by −1. Using these operations, any matrix can be transformed to a lower (or upper) triangular matrix, and for such matrices the determinant equals the product of the entries on the main diagonal; this provides a method to calculate the determinant of any matrix. Finally, the Laplace expansionLaplace expansionIn linear algebra, the Laplace expansion, named after PierreSimon Laplace, also called cofactor expansion, is an expression for the determinant B of...
expresses the determinant in terms of minorsMinor (linear algebra)In linear algebra, a minor of a matrix A is the determinant of some smaller square matrix, cut down from A by removing one or more of its rows or columns...
, i.e., determinants of smaller matrices. This expansion can be used for a recursive definition of determinants (taking as starting case the determinant of a 1by1 matrix, which is its unique entry, or even the determinant of a 0by0 matrix, which is 1), that can be seen to be equivalent to the Leibniz formula. Determinants can be used to solve linear systemLinear systemA linear system is a mathematical model of a system based on the use of a linear operator.Linear systems typically exhibit features and properties that are much simpler than the general, nonlinear case....
s using Cramer's ruleCramer's ruleIn linear algebra, Cramer's rule is a theorem, which gives an expression for the solution of a system of linear equations with as many equations as unknowns, valid in those cases where there is a unique solution...
, where the division of the determinants of two related square matrices equates to the value of each of the system's variables.
Eigenvalues and eigenvectors
A number λ and a nonzero vector v satisfying Av = λv
are called an eigenvalue and an eigenvector of A, respectively.Eigen means "own" in GermanGerman languageGerman is a West Germanic language, related to and classified alongside English and Dutch. With an estimated 90 – 98 million native speakers, German is one of the world's major languages and is the most widelyspoken first language in the European Union....
and in DutchDutch languageDutch is a West Germanic language and the native language of the majority of the population of the Netherlands, Belgium, and Suriname, the three member states of the Dutch Language Union. Most speakers live in the European Union, where it is a first language for about 23 million and a second...
. The number λ is an eigenvalue of an n×nmatrix A if and only if A−λI_{n} is not invertible, which is equivalentLogical equivalenceIn logic, statements p and q are logically equivalent if they have the same logical content.Syntactically, p and q are equivalent if each can be proved from the other...
to
The function p_{}A(t) = det(A−tI) is called the characteristic polynomialCharacteristic polynomialIn linear algebra, one associates a polynomial to every square matrix: its characteristic polynomial. This polynomial encodes several important properties of the matrix, most notably its eigenvalues, its determinant and its trace....
of A, its degreeDegree of a polynomialThe degree of a polynomial represents the highest degree of a polynominal's terms , should the polynomial be expressed in canonical form . The degree of an individual term is the sum of the exponents acting on the term's variables...
is n. Therefore p_{A}(t) has at most n different roots, i.e., eigenvalues of the matrix. They may be complex even if the entries of A are real. According to the Cayley–Hamilton theoremCayley–Hamilton theoremIn linear algebra, the Cayley–Hamilton theorem states that every square matrix over a commutative ring satisfies its own characteristic equation....
, p_{A}(A) = 0, that is to say, the characteristic polynomial applied to the matrix itself yields the zero matrix.
Symmetry
A square matrix A that is equal to its transpose, i.e., A = A^{T}, is a symmetric matrix. If instead, A was equal to the negative of its transpose, i.e., A = −A^{T}, then A is a skewsymmetric matrixSkewsymmetric matrixIn mathematics, and in particular linear algebra, a skewsymmetric matrix is a square matrix A whose transpose is also its negative; that is, it satisfies the equation If the entry in the and is aij, i.e...
. In complex matrices, symmetry is often replaced by the concept of Hermitian matrices, which satisfy A^{∗} = A, where the star or asteriskAsteriskAn asterisk is a typographical symbol or glyph. It is so called because it resembles a conventional image of a star. Computer scientists and mathematicians often pronounce it as star...
denotes the conjugate transposeConjugate transposeIn mathematics, the conjugate transpose, Hermitian transpose, Hermitian conjugate, or adjoint matrix of an mbyn matrix A with complex entries is the nbym matrix A* obtained from A by taking the transpose and then taking the complex conjugate of each entry...
of the matrix, i.e., the transpose of the complex conjugateComplex conjugateIn mathematics, complex conjugates are a pair of complex numbers, both having the same real part, but with imaginary parts of equal magnitude and opposite signs...
of A.
By the spectral theoremSpectral theoremIn mathematics, particularly linear algebra and functional analysis, the spectral theorem is any of a number of results about linear operators or about matrices. In broad terms the spectral theorem provides conditions under which an operator or a matrix can be diagonalized...
, real symmetric matrices and complex Hermitian matrices have an eigenbasis; i.e., every vector is expressible as a linear combinationLinear combinationIn mathematics, a linear combination is an expression constructed from a set of terms by multiplying each term by a constant and adding the results...
of eigenvectors. In both cases, all eigenvalues are real. This theorem can be generalized to infinitedimensional situations related to matrices with infinitely many rows and columns, see below.
Definiteness
Matrix A; definiteness; associated quadratic form Q_{A}(x,y);
set of vectors (x,y) such that Q_{A}(x,y)=1positive definite indefinite 1/4 x^{2} + 1/4y^{2} 1/4 x^{2} − 1/4 y^{2}
EllipseEllipseIn geometry, an ellipse is a plane curve that results from the intersection of a cone by a plane in a way that produces a closed curve. Circles are special cases of ellipses, obtained when the cutting plane is orthogonal to the cone's axis...
HyperbolaHyperbolaIn mathematics a hyperbola is a curve, specifically a smooth curve that lies in a plane, which can be defined either by its geometric properties or by the kinds of equations for which it is the solution set. A hyperbola has two pieces, called connected components or branches, which are mirror...
A symmetric n×nmatrix is called positivedefinitePositivedefinite matrixIn linear algebra, a positivedefinite matrix is a matrix that in many ways is analogous to a positive real number. The notion is closely related to a positivedefinite symmetric bilinear form ....
(respectively negativedefinite; indefinite), if for all nonzero vectors x ∈ R^{n} the associated quadratic formQuadratic formIn mathematics, a quadratic form is a homogeneous polynomial of degree two in a number of variables. For example,4x^2 + 2xy  3y^2\,\!is a quadratic form in the variables x and y....
given by Q(x) = x^{T}Ax
takes only positive values (respectively only negative values; both some negative and some positive values). If the quadratic form takes only nonnegative (respectively only nonpositive) values, the symmetric matrix is called positivesemidefinite (respectively negativesemidefinite); hence the matrix is indefinite precisely when it is neither positivesemidefinite nor negativesemidefinite.
A symmetric matrix is positivedefinite if and only if all its eigenvalues are positive. The table at the right shows two possibilities for 2by2 matrices.
Allowing as input two different vectors instead yields the bilinear form associated to A: B_{A} (x, y) = x^{T}Ay.
Computational aspects
In addition to theoretical knowledge of properties of matrices and their relation to other fields, it is important for practical purposes to perform matrix calculations effectively and precisely. The domain studying these matters is called numerical linear algebraNumerical linear algebraNumerical linear algebra is the study of algorithms for performing linear algebra computations, most notably matrix operations, on computers. It is often a fundamental part of engineering and computational science problems, such as image and signal processing, Telecommunication, computational...
. As with other numerical situations, two main aspects are the complexity of algorithms and their numerical stabilityNumerical stabilityIn the mathematical subfield of numerical analysis, numerical stability is a desirable property of numerical algorithms. The precise definition of stability depends on the context, but it is related to the accuracy of the algorithm....
. Many problems can be solved by both direct algorithms or iterative approaches. For example, finding eigenvectors can be done by finding a sequence of vectors x_{n} convergingLimit of a sequenceThe limit of a sequence is, intuitively, the unique number or point L such that the terms of the sequence become arbitrarily close to L for "large" values of n...
to an eigenvector when n tends to infinityInfinityInfinity is a concept in many fields, most predominantly mathematics and physics, that refers to a quantity without bound or end. People have developed various ideas throughout history about the nature of infinity...
.
Determining the complexity of an algorithm means finding upper boundUpper boundIn mathematics, especially in order theory, an upper bound of a subset S of some partially ordered set is an element of P which is greater than or equal to every element of S. The term lower bound is defined dually as an element of P which is lesser than or equal to every element of S...
s or estimates of how many elementary operations such as additions and multiplications of scalars are necessary to perform some algorithm, e.g., multiplication of matrices. For example, calculating the matrix product of two nbyn matrix using the definition given above needs n^{3} multiplications, since for any of the n^{2} entries of the product, n multiplications are necessary. The Strassen algorithmStrassen algorithmIn the mathematical discipline of linear algebra, the Strassen algorithm, named after Volker Strassen, is an algorithm used for matrix multiplication...
outperforms this "naive" algorithm; it needs only n^{2.807} multiplications. A refined approach also incorporates specific features of the computing devices.
In many practical situations additional information about the matrices involved is known. An important case are sparse matricesSparse matrixIn the subfield of numerical analysis, a sparse matrix is a matrix populated primarily with zeros . The term itself was coined by Harry M. Markowitz....
, i.e., matrices most of whose entries are zero. There are specifically adapted algorithms for, say, solving linear systems Ax = b for sparse matrices A, such as the conjugate gradient methodConjugate gradient methodIn mathematics, the conjugate gradient method is an algorithm for the numerical solution of particular systems of linear equations, namely those whose matrix is symmetric and positivedefinite. The conjugate gradient method is an iterative method, so it can be applied to sparse systems that are too...
.
An algorithm is, roughly speaking, numerically stable, if little deviations (such as rounding errors) do not lead to big deviations in the result. For example, calculating the inverse of a matrix via Laplace's formula (Adj (A) denotes the adjugate matrixAdjugate matrixIn linear algebra, the adjugate or classical adjoint of a square matrix is a matrix that plays a role similar to the inverse of a matrix; it can however be defined for any square matrix without the need to perform any divisions....
of A) A^{−1} = Adj(A) / det(A)
may lead to significant rounding errors if the determinant of the matrix is very small. The norm of a matrix can be used to capture the conditioningCondition numberIn the field of numerical analysis, the condition number of a function with respect to an argument measures the asymptotically worst case of how much the function can change in proportion to small changes in the argument...
of linear algebraic problems, such as computing a matrix' inverse.
Although most computer languages are not designed with commands or libraries for matrices, as early as the 1970s, some engineering desktop computers such as the HP 9830HP 9830The HP 9800 was a family of what were initially called programmable calculators and later desktop computers made by HewlettPackard, replacing their first HP 9100 calculator...
had ROM cartridges to add BASIC commands for matrices. Some computer languages such as APL were designed to manipulate matrices, and various mathematical programs can be used to aid computing with matrices.
Matrix decomposition methods
There are several methods to render matrices into a more easily accessible form. They are generally referred to as matrix transformation or matrix decomposition techniques. The interest of all these decomposition techniques is that they preserve certain properties of the matrices in question, such as determinant, rank or inverse, so that these quantities can be calculated after applying the transformation, or that certain matrix operations are algorithmically easier to carry out for some types of matrices.
The LU decompositionLU decompositionIn linear algebra, LU decomposition is a matrix decomposition which writes a matrix as the product of a lower triangular matrix and an upper triangular matrix. The product sometimes includes a permutation matrix as well. This decomposition is used in numerical analysis to solve systems of linear...
factors matrices as a product of lower (L) and an upper triangular matricesTriangular matrixIn the mathematical discipline of linear algebra, a triangular matrix is a special kind of square matrix where either all the entries below or all the entries above the main diagonal are zero...
(U). Once this decomposition is calculated, linear systems can be solved more efficiently, by a simple technique called forward and back substitution. Likewise, inverses of triangular matrices are algorithmically easier to calculate. The Gaussian elimination is a similar algorithm; it transforms any matrix to row echelon formRow echelon formIn linear algebra a matrix is in row echelon form if* All nonzero rows are above any rows of all zeroes, and...
. Both methods proceed by multiplying the matrix by suitable elementary matrices, which correspond to permuting rows or columnsPermutation matrixIn mathematics, in matrix theory, a permutation matrix is a square binary matrix that has exactly one entry 1 in each row and each column and 0s elsewhere...
and adding multiples of one row to another row. Singular value decompositionSingular value decompositionIn linear algebra, the singular value decomposition is a factorization of a real or complex matrix, with many useful applications in signal processing and statistics....
expresses any matrix A as a product UDV^{∗}, where U and V are unitary matrices and D is a diagonal matrix.
The eigendecomposition or diagonalization expresses A as a product VDV^{−1}, where D is a diagonal matrix and V is a suitable invertible matrix. If A can be written in this form, it is called diagonalizableDiagonalizable matrixIn linear algebra, a square matrix A is called diagonalizable if it is similar to a diagonal matrix, i.e., if there exists an invertible matrix P such that P −1AP is a diagonal matrix...
. More generally, and applicable to all matrices, the Jordan decomposition transforms a matrix into Jordan normal formJordan normal formIn linear algebra, a Jordan normal form of a linear operator on a finitedimensional vector space is an upper triangular matrix of a particular form called Jordan matrix, representing the operator on some basis...
, that is to say matrices whose only nonzero entries are the eigenvalues λ_{1} to λ_{n} of A, placed on the main diagonal and possibly entries equal to one directly above the main diagonal, as shown at the right. Given the eigendecomposition, the n^{th} power of A (i.e., nfold iterated matrix multiplication) can be calculated via A^{n} = (VDV^{−1})^{n} = VDV^{−1}VDV^{−1}...VDV^{−1} = VD^{n}V^{−1}
and the power of a diagonal matrix can be calculated by taking the corresponding powers of the diagonal entries, which is much easier than doing the exponentiation for A instead. This can be used to compute the matrix exponentialMatrix exponentialIn mathematics, the matrix exponential is a matrix function on square matrices analogous to the ordinary exponential function. Abstractly, the matrix exponential gives the connection between a matrix Lie algebra and the corresponding Lie group....
e^{A}, a need frequently arising in solving linear differential equationLinear differential equationLinear differential equations are of the formwhere the differential operator L is a linear operator, y is the unknown function , and the right hand side ƒ is a given function of the same nature as y...
s, matrix logarithms and square roots of matricesSquare root of a matrixIn mathematics, the square root of a matrix extends the notion of square root from numbers to matrices. A matrix B is said to be a square root of A if the matrix product B · B is equal to A.Properties:...
. To avoid numerically illconditionedCondition numberIn the field of numerical analysis, the condition number of a function with respect to an argument measures the asymptotically worst case of how much the function can change in proportion to small changes in the argument...
situations, further algorithms such as the Schur decompositionSchur decompositionIn the mathematical discipline of linear algebra, the Schur decomposition or Schur triangulation, named after Issai Schur, is a matrix decomposition. Statement :...
can be employed.
Abstract algebraic aspects and generalizations
Matrices can be generalized in different ways. Abstract algebra uses matrices with entries in more general fieldsField (mathematics)In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...
or even ringsRing (mathematics)In mathematics, a ring is an algebraic structure consisting of a set together with two binary operations usually called addition and multiplication, where the set is an abelian group under addition and a semigroup under multiplication such that multiplication distributes over addition...
, while linear algebra codifies properties of matrices in the notion of linear maps. It is possible to consider matrices with infinitely many columns and rows. Another extension are tensors, which can be seen as higherdimensional arrays of numbers, as opposed to vectors, which can often be realised as sequences of numbers, while matrices are rectangular or twodimensional array of numbers. Matrices, subject to certain requirements tend to form groupsGroup (mathematics)In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...
known as matrix groups.
Matrices with more general entries
This article focuses on matrices whose entries are real or complex numberComplex numberA complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the onedimensional number line to the twodimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...
s. However, matrices can be considered with much more general types of entries than real or complex numbers. As a first step of generalization, any fieldField (mathematics)In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...
, i.e., a set where additionAdditionAddition is a mathematical operation that represents combining collections of objects together into a larger collection. It is signified by the plus sign . For example, in the picture on the right, there are 3 + 2 apples—meaning three apples and two other apples—which is the same as five apples....
, subtractionSubtractionIn arithmetic, subtraction is one of the four basic binary operations; it is the inverse of addition, meaning that if we start with any number and add any number and then subtract the same number we added, we return to the number we started with...
, multiplicationMultiplicationMultiplication is the mathematical operation of scaling one number by another. It is one of the four basic operations in elementary arithmetic ....
and divisionDivision (mathematics)rightthumb200px20 \div 4=5In mathematics, especially in elementary arithmetic, division is an arithmetic operation.Specifically, if c times b equals a, written:c \times b = a\,...
operations are defined and wellbehaved, may be used instead of R or C, for example rational numberRational numberIn mathematics, a rational number is any number that can be expressed as the quotient or fraction a/b of two integers, with the denominator b not equal to zero. Since b may be equal to 1, every integer is a rational number...
s or finite fieldFinite fieldIn abstract algebra, a finite field or Galois field is a field that contains a finite number of elements. Finite fields are important in number theory, algebraic geometry, Galois theory, cryptography, and coding theory...
s. For example, coding theoryCoding theoryCoding theory is the study of the properties of codes and their fitness for a specific application. Codes are used for data compression, cryptography, errorcorrection and more recently also for network coding...
makes use of matrices over finite fields. Wherever eigenvalues are considered, as these are roots of a polynomial they may exist only in a larger field than that of the coefficients of the matrix; for instance they may be complex in case of a matrix with real entries. The possibility to reinterpret the entries of a matrix as elements of a larger field (e.g., to view a real matrix as a complex matrix whose entries happen to be all real) then allows considering each square matrix to possess a full set of eigenvalues. Alternatively one can consider only matrices with entries in an algebraically closed fieldAlgebraically closed fieldIn mathematics, a field F is said to be algebraically closed if every polynomial with one variable of degree at least 1, with coefficients in F, has a root in F.Examples:...
, such as C, from the outset.
More generally, abstract algebra makes great use of matrices with entries in a ringRing (mathematics)In mathematics, a ring is an algebraic structure consisting of a set together with two binary operations usually called addition and multiplication, where the set is an abelian group under addition and a semigroup under multiplication such that multiplication distributes over addition...
R. Rings are a more general notion than fields in that no division operation exists. The very same addition and multiplication operations of matrices extend to this setting, too. The set M(n, R) of all square nbyn matrices over R is a ring called matrix ringMatrix ringIn abstract algebra, a matrix ring is any collection of matrices forming a ring under matrix addition and matrix multiplication. The set of n×n matrices with entries from another ring is a matrix ring, as well as some subsets of infinite matrices which form infinite matrix rings...
, isomorphic to the endomorphism ringEndomorphism ringIn abstract algebra, one associates to certain objects a ring, the object's endomorphism ring, which encodes several internal properties of the object; this may be denoted End...
of the left RmoduleModule (mathematics)In abstract algebra, the concept of a module over a ring is a generalization of the notion of vector space, wherein the corresponding scalars are allowed to lie in an arbitrary ring...
R^{n}. If the ring R is commutativeCommutative ringIn ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra....
, i.e., its multiplication is commutative, then M(n, R) is a unitary noncommutative (unless n = 1) associative algebraAssociative algebraIn mathematics, an associative algebra A is an associative ring that has a compatible structure of a vector space over a certain field K or, more generally, of a module over a commutative ring R...
over R. The determinantDeterminantIn linear algebra, the determinant is a value associated with a square matrix. It can be computed from the entries of the matrix by a specific arithmetic expression, while other ways to determine its value exist as well...
of square matrices over a commutative ring R can still be defined using the Leibniz formula; such a matrix is invertible if and only if its determinant is invertible in R, generalising the situation over a field F, where every nonzero element is invertible. Matrices over superrings are called supermatricesSupermatrixIn mathematics and theoretical physics, a supermatrix is a Z2graded analog of an ordinary matrix. Specifically, a supermatrix is a 2×2 block matrix with entries in a superalgebra...
.
Matrices do not always have all their entries in the same ring – or even in any ring at all. One special but common case is block matricesBlock matrixIn the mathematical discipline of matrix theory, a block matrix or a partitioned matrix is a matrix broken into sections called blocks. Looking at it another way, the matrix is written in terms of smaller matrices. We group the rows and columns into adjacent 'bunches'. A partition is the rectangle...
, which may be considered as matrices whose entries themselves are matrices. The entries need not be quadratic matrices, and thus need not be members of any ordinary ringRing (mathematics)In mathematics, a ring is an algebraic structure consisting of a set together with two binary operations usually called addition and multiplication, where the set is an abelian group under addition and a semigroup under multiplication such that multiplication distributes over addition...
; but their sizes must fulfil certain compatibility conditions.
Relationship to linear maps
Linear maps R^{n} → R^{m} are equivalent to mbyn matrices, as described above. More generally, any linear map between finitedimensional vector spaceVector spaceA vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...
s can be described by a matrix A = (a_{ij}), after choosing basesBasis (linear algebra)In linear algebra, a basis is a set of linearly independent vectors that, in a linear combination, can represent every vector in a given vector space or free module, or, more simply put, which define a "coordinate system"...
v_{1}, ..., v_{n} of V, and w_{1}, ..., w_{m} of W (so n is the dimension of V and m is the dimension of W), which is such that
In other words, column j of A expresses the image of v_{j} in terms of the basis vectors w_{i} of W; thus this relation uniquely determines the entries of the matrix A. Note that the matrix depends on the choice of the bases: different choices of bases give rise to different, but equivalent matrices. Many of the above concrete notions can be reinterpreted in this light, for example, the transpose matrix A^{T} describes the transpose of the linear map given by A, with respect to the dual basesDual spaceIn mathematics, any vector space, V, has a corresponding dual vector space consisting of all linear functionals on V. Dual vector spaces defined on finitedimensional vector spaces can be used for defining tensors which are studied in tensor algebra...
.
More generally, the set of m×n matrices can be used to represent the Rlinear maps between the free modules R^{m} and R^{n} for an arbitrary ring R with unity. When n = m composition of these maps is possible, and this gives rise to the matrix ringMatrix ringIn abstract algebra, a matrix ring is any collection of matrices forming a ring under matrix addition and matrix multiplication. The set of n×n matrices with entries from another ring is a matrix ring, as well as some subsets of infinite matrices which form infinite matrix rings...
of n×n matrices representing the endomorphism ringEndomorphism ringIn abstract algebra, one associates to certain objects a ring, the object's endomorphism ring, which encodes several internal properties of the object; this may be denoted End...
of R^{n}.
Matrix groups
A groupGroup (mathematics)In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...
is a mathematical structure consisting of a set of objects together with a binary operationBinary operationIn mathematics, a binary operation is a calculation involving two operands, in other words, an operation whose arity is two. Examples include the familiar arithmetic operations of addition, subtraction, multiplication and division....
, i.e., an operation combining any two objects to a third, subject to certain requirements. A group in which the objects are matrices and the group operation is matrix multiplication is called a matrix group.Additionally, the group is required to be closed in the general linear group. Since in a group every element has to be invertible, the most general matrix groups are the groups of all invertible matrices of a given size, called the general linear groupGeneral linear groupIn mathematics, the general linear group of degree n is the set of n×n invertible matrices, together with the operation of ordinary matrix multiplication. This forms a group, because the product of two invertible matrices is again invertible, and the inverse of an invertible matrix is invertible...
s.
Any property of matrices that is preserved under matrix products and inverses can be used to define further matrix groups. For example, matrices with a given size and with a determinant of 1 form a subgroupSubgroupIn group theory, given a group G under a binary operation *, a subset H of G is called a subgroup of G if H also forms a group under the operation *. More precisely, H is a subgroup of G if the restriction of * to H x H is a group operation on H...
of (i.e., a smaller group contained in) their general linear group, called a special linear groupSpecial linear groupIn mathematics, the special linear group of degree n over a field F is the set of n×n matrices with determinant 1, with the group operations of ordinary matrix multiplication and matrix inversion....
. Orthogonal matricesOrthogonal matrixIn linear algebra, an orthogonal matrix , is a square matrix with real entries whose columns and rows are orthogonal unit vectors ....
, determined by the condition M^{T}M = I,
form the orthogonal groupOrthogonal groupIn mathematics, the orthogonal group of degree n over a field F is the group of n × n orthogonal matrices with entries from F, with the group operation of matrix multiplication...
. They are called orthogonal since the associated linear transformations of R^{n} preserve angles in the sense that the scalar product of two vectors is unchanged after applying M to them: · (Mw) = v · w.
Every finite groupFinite groupIn mathematics and abstract algebra, a finite group is a group whose underlying set G has finitely many elements. During the twentieth century, mathematicians investigated certain aspects of the theory of finite groups in great depth, especially the local theory of finite groups, and the theory of...
is isomorphic to a matrix group, as one can see by considering the regular representationRegular representationIn mathematics, and in particular the theory of group representations, the regular representation of a group G is the linear representation afforded by the group action of G on itself by translation....
of the symmetric groupSymmetric groupIn mathematics, the symmetric group Sn on a finite set of n symbols is the group whose elements are all the permutations of the n symbols, and whose group operation is the composition of such permutations, which are treated as bijective functions from the set of symbols to itself...
. General groups can be studied using matrix groups, which are comparatively wellunderstood, by means of representation theoryRepresentation theoryRepresentation theory is a branch of mathematics that studies abstract algebraic structures by representing their elements as linear transformations of vector spaces, and studiesmodules over these abstract algebraic structures...
.
Infinite matrices
It is also possible to consider matrices with infinitely many rows and/or columns even if, being infinite objects, one cannot write down such matrices explicitly. All that matters is that for every element in the set indexing rows, and every element in the set indexing columns, there is a welldefined entry (these index sets need not even be subsets of the natural numbers). The basic operations of addition, subtraction, scalar multiplication and transposition can still be defined without problem; however matrix multiplication may involve infinite summations to define the resulting entries, and these are not defined in general.
If R is any ring with unity, then the ring of endomorphisms of as a right R module is isomorphic to the ring of column finite matrices whose entries are indexed by , and whose columns each contain only finitely many nonzero entries. The endomorphisms of M considered as a left R module result in an analogous object, the row finite matrices whose rows each only have finitely many nonzero entries.
If infinite matrices are used to describe linear maps, then only those matrices can be used all of whose columns have but a finite number of nonzero entries, for the following reason. For a matrix A to describe a linear map f: V→W, bases for both spaces must have been chosen; recall that by definition this means that every vector in the space can be written uniquely as a (finite) linear combinationLinear combinationIn mathematics, a linear combination is an expression constructed from a set of terms by multiplying each term by a constant and adding the results...
of basis vectors, so that written as a (column) vector v of coefficients, only finitely many entries v_{i} are nonzero. Now the columns of A describe the images by f of individual basis vectors of V in the basis of W, which is only meaningful if these columns have only finitely many nonzero entries. There is no restriction on the rows of A however: in the product A·v there are only finitely many nonzero coefficients of v involved, so every one of its entries, even if it is given as an infinite sum of products, involves only finitely many nonzero terms and is therefore well defined. Moreover this amounts to forming a linear combination of the columns of A that effectively involves only finitely many of them, whence the result has only finitely many nonzero entries, because each of those columns do. One also sees that products of two matrices of the given type is well defined (provided as usual that the columnindex and rowindex sets match), is again of the same type, and corresponds to the composition of linear maps.
If R is a normed ring, then the condition of row or column finiteness can be relaxed. With the norm in place, absolutely convergent series can be used instead of finite sums. For example, the matrices whose column sums are absolutely convergent sequences form a ring. Analogously of course, the matrices whose row sums are absolutely convergent series also form a ring.
In that vein, infinite matrices can also be used to describe operators on Hilbert spaces, where convergence and continuityContinuous functionIn mathematics, a continuous function is a function for which, intuitively, "small" changes in the input result in "small" changes in the output. Otherwise, a function is said to be "discontinuous". A continuous function with a continuous inverse function is called "bicontinuous".Continuity of...
questions arise, which again results in certain constraints that have to be imposed. However, the explicit point of view of matrices tends to obfuscate the matter,"Not much of matrix theory carries over to infinitedimensional spaces, and what does is not so useful, but it sometimes helps." and the abstract and more powerful tools of functional analysisFunctional analysisFunctional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limitrelated structure and the linear operators acting upon these spaces and respecting these structures in a suitable sense...
can be used instead.
Empty matrices
An empty matrix is a matrix in which the number of rows or columns (or both) is zero. Empty matrices help dealing with maps involving the zero vector space. For example, if A is a 3by0 matrix A and B is a 0by3 matrix, then AB is the 3by3 zero matrix corresponding to the null map from a 3dimensional space V to itself, while BA is a 0by0 matrix. There is no common notation for empty matrices, but most computer algebra systemComputer algebra systemA computer algebra system is a software program that facilitates symbolic mathematics. The core functionality of a CAS is manipulation of mathematical expressions in symbolic form.Symbolic manipulations:...
s allow creating and computing with them. The determinant of the 0by0 matrix is 1 as follows from regarding the empty productEmpty productIn mathematics, an empty product, or nullary product, is the result of multiplying no factors. It is equal to the multiplicative identity 1, given that it exists for the multiplication operation in question, just as the empty sum—the result of adding no numbers—is zero, or the additive...
occurring in the Leibniz formula for the determinant as 1. This value is also consistent with the fact that the identity map from any finite dimensional space to itself has determinant 1, a fact that is often used as a part of the characterization of determinants.
Applications
There are numerous applications of matrices, both in mathematics and other sciences. Some of them merely take advantage of the compact representation of a set of numbers in a matrix. For example, in game theoryGame theoryGame theory is a mathematical method for analyzing calculated circumstances, such as in games, where a person’s success is based upon the choices of others...
and economicsEconomicsEconomics is the social science that analyzes the production, distribution, and consumption of goods and services. The term economics comes from the Ancient Greek from + , hence "rules of the house"...
, the payoff matrix encodes the payoff for two players, depending on which out of a given (finite) set of alternatives the players choose. Text miningText miningText mining, sometimes alternately referred to as text data mining, roughly equivalent to text analytics, refers to the process of deriving highquality information from text. Highquality information is typically derived through the devising of patterns and trends through means such as...
and automated thesaurusThesaurusA thesaurus is a reference work that lists words grouped together according to similarity of meaning , in contrast to a dictionary, which contains definitions and pronunciations...
compilation makes use of documentterm matricesDocumentterm matrixA documentterm matrix or termdocument matrix is a mathematical matrix that describes the frequency of terms that occur in a collection of documents. In a documentterm matrix, rows correspond to documents in the collection and columns correspond to terms. There are various schemes for determining...
such as tfidf to track frequencies of certain words in several documents.
Complex numbers can be represented by particular real 2by2 matrices via
under which addition and multiplication of complex numbers and matrices correspond to each other. For example, 2by2 rotation matrices represent the multiplication with some complex number of absolute valueAbsolute valueIn mathematics, the absolute value a of a real number a is the numerical value of a without regard to its sign. So, for example, the absolute value of 3 is 3, and the absolute value of 3 is also 3...
1, as above. A similar interpretation is possible for quaternionQuaternionIn mathematics, the quaternions are a number system that extends the complex numbers. They were first described by Irish mathematician Sir William Rowan Hamilton in 1843 and applied to mechanics in threedimensional space...
s.
Early encryptionEncryptionIn cryptography, encryption is the process of transforming information using an algorithm to make it unreadable to anyone except those possessing special knowledge, usually referred to as a key. The result of the process is encrypted information...
techniques such as the Hill cipherHill cipherIn classical cryptography, the Hill cipher is a polygraphic substitution cipher based on linear algebra. Invented by Lester S. Hill in 1929, it was the first polygraphic cipher in which it was practical to operate on more than three symbols at once. The following discussion assumes an elementary...
also used matrices. However, due to the linear nature of matrices, these codes are comparatively easy to break. Computer graphicsComputer graphicsComputer graphics are graphics created using computers and, more generally, the representation and manipulation of image data by a computer with help from specialized software and hardware....
uses matrices both to represent objects and to calculate transformations of objects using affine rotation matrices to accomplish tasks such as projecting a threedimensional object onto a twodimensional screen, corresponding to a theoretical camera observation. Matrices over a polynomial ringPolynomial ringIn mathematics, especially in the field of abstract algebra, a polynomial ring is a ring formed from the set of polynomials in one or more variables with coefficients in another ring. Polynomial rings have influenced much of mathematics, from the Hilbert basis theorem, to the construction of...
are important in the study of control theoryControl theoryControl theory is an interdisciplinary branch of engineering and mathematics that deals with the behavior of dynamical systems. The desired output of a system is called the reference...
.
ChemistryChemistryChemistry is the science of matter, especially its chemical reactions, but also its composition, structure and properties. Chemistry is concerned with atoms and their interactions with other atoms, and particularly with the properties of chemical bonds....
makes use of matrices in various ways, particularly since the use of quantum theoryQuantum mechanicsQuantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particlelike and wavelike behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...
to discuss molecular bondingChemical bondA chemical bond is an attraction between atoms that allows the formation of chemical substances that contain two or more atoms. The bond is caused by the electromagnetic force attraction between opposite charges, either between electrons and nuclei, or as the result of a dipole attraction...
and spectroscopySpectroscopySpectroscopy is the study of the interaction between matter and radiated energy. Historically, spectroscopy originated through the study of visible light dispersed according to its wavelength, e.g., by a prism. Later the concept was expanded greatly to comprise any interaction with radiative...
. Examples are the overlap matrixOverlap matrixThe overlap matrix is a square matrix, used in quantum chemistry to describe the interrelationship of a set of basis vectors of a quantum system. In particular, if the vectors are orthogonal to one another, the overlap matrix will be diagonal. In addition, if the basis vectors form an...
and the Fock matrixFock matrixIn the HartreeFock method of quantum mechanics, the Fock matrix is a matrix approximating the singleelectron energy operator of a given quantum system in a given set of basis vectors....
used in solving the Roothaan equationsRoothaan equationsThe Roothaan equations are a representation of the HartreeFock equation in a non orthonormal basis set which can be of Gaussiantype or Slatertype. It applies to closedshell molecules or atoms where all molecular orbitals or atomic orbitals, respectively, are doubly occupied. This is generally...
to obtain the molecular orbitalMolecular orbitalIn chemistry, a molecular orbital is a mathematical function describing the wavelike behavior of an electron in a molecule. This function can be used to calculate chemical and physical properties such as the probability of finding an electron in any specific region. The term "orbital" was first...
s of the Hartree–Fock method.
Graph theory
The adjacency matrixAdjacency matrixIn mathematics and computer science, an adjacency matrix is a means of representing which vertices of a graph are adjacent to which other vertices...
of a finite graph is a basic notion of graph theoryGraph theoryIn mathematics and computer science, graph theory is the study of graphs, mathematical structures used to model pairwise relations between objects from a certain collection. A "graph" in this context refers to a collection of vertices or 'nodes' and a collection of edges that connect pairs of...
. It saves which vertices of the graph are connected by an edge. Matrices containing just two different values (0 and 1 meaning for example "yes" and "no") are called logical matrices. The distance (or cost) matrixDistance matrixIn mathematics, computer science and graph theory, a distance matrix is a matrix containing the distances, taken pairwise, of a set of points...
contains information about distances of the edges. These concepts can be applied to websiteWebsiteA website, also written as Web site, web site, or simply site, is a collection of related web pages containing images, videos or other digital assets. A website is hosted on at least one web server, accessible via a network such as the Internet or a private local area network through an Internet...
s connected hyperlinkHyperlinkIn computing, a hyperlink is a reference to data that the reader can directly follow, or that is followed automatically. A hyperlink points to a whole document or to a specific element within a document. Hypertext is text with hyperlinks...
s or cities connected by roads etc., in which case (unless the road network is extremely dense) the matrices tend to be sparseSparse matrixIn the subfield of numerical analysis, a sparse matrix is a matrix populated primarily with zeros . The term itself was coined by Harry M. Markowitz....
, i.e., contain few nonzero entries. Therefore, specifically tailored matrix algorithms can be used in network theoryNetwork theoryNetwork theory is an area of computer science and network science and part of graph theory. It has application in many disciplines including statistical physics, particle physics, computer science, biology, economics, operations research, and sociology...
.
Analysis and geometry
The Hessian matrixHessian matrixIn mathematics, the Hessian matrix is the square matrix of secondorder partial derivatives of a function; that is, it describes the local curvature of a function of many variables. The Hessian matrix was developed in the 19th century by the German mathematician Ludwig Otto Hesse and later named...
of a differentiable functionDifferentiable functionIn calculus , a differentiable function is a function whose derivative exists at each point in its domain. The graph of a differentiable function must have a nonvertical tangent line at each point in its domain...
ƒ: R^{n} → R consists of the second derivativeSecond derivativeIn calculus, the second derivative of a function ƒ is the derivative of the derivative of ƒ. Roughly speaking, the second derivative measures how the rate of change of a quantity is itself changing; for example, the second derivative of the position of a vehicle with respect to time is...
s of ƒ with respect to the several coordinate directions, i.e.
It encodes information about the local growth behaviour of the function: given a critical pointCritical point (mathematics)In calculus, a critical point of a function of a real variable is any value in the domain where either the function is not differentiable or its derivative is 0. The value of the function at a critical point is a critical value of the function...
x = (x_{1}, ..., x_{n}), i.e., a point where the first partial derivatives of ƒ vanish, the function has a local minimum if the Hessian matrix is positive definite. Quadratic programmingQuadratic programmingQuadratic programming is a special type of mathematical optimization problem. It is the problem of optimizing a quadratic function of several variables subject to linear constraints on these variables....
can be used to find global minima or maxima of quadratic functions closely related to the ones attached to matrices (see above).
Another matrix frequently used in geometrical situations is the Jacobi matrix of a differentiable map f: R^{n} → R^{m}. If f_{1}, ..., f_{m} denote the components of f, then the Jacobi matrix is defined as
If n > m, and if the rank of the Jacobi matrix attains its maximal value m, f is locally invertible at that point, by the implicit function theoremImplicit function theoremIn multivariable calculus, the implicit function theorem is a tool which allows relations to be converted to functions. It does this by representing the relation as the graph of a function. There may not be a single function whose graph is the entire relation, but there may be such a function on...
.
Partial differential equationPartial differential equationIn mathematics, partial differential equations are a type of differential equation, i.e., a relation involving an unknown function of several independent variables and their partial derivatives with respect to those variables...
s can be classified by considering the matrix of coefficients of the highestorder differential operators of the equation. For elliptic partial differential equations this matrix is positive definite, which has decisive influence on the set of possible solutions of the equation in question.
The finite element methodFinite element methodThe finite element method is a numerical technique for finding approximate solutions of partial differential equations as well as integral equations...
is an important numerical method to solve partial differential equations, widely applied in simulating complex physical systems. It attempts to approximate the solution to some equation by piecewise linear functions, where the pieces are chosen with respect to a sufficiently fine grid, which in turn can be recast as a matrix equation.
Probability theory and statistics
Stochastic matricesStochastic matrixIn mathematics, a stochastic matrix is a matrix used to describe the transitions of a Markov chain. It has found use in probability theory, statistics and linear algebra, as well as computer science...
are square matrices whose rows are probability vectorProbability vectorStochastic vector redirects here. For the concept of a random vector, see Multivariate random variable.In mathematics and statistics, a probability vector or stochastic vector is a vector with nonnegative entries that add up to one....
s, i.e., whose entries sum up to one. Stochastic matrices are used to define Markov chainMarkov chainA Markov chain, named after Andrey Markov, is a mathematical system that undergoes transitions from one state to another, between a finite or countable number of possible states. It is a random process characterized as memoryless: the next state depends only on the current state and not on the...
s with finitely many states. A row of the stochastic matrix gives the probability distribution for the next position of some particle currently in the state that corresponds to the row. Properties of the Markov chain like absorbing states, i.e., states that any particle attains eventually, can be read off the eigenvectors of the transition matrices.
Statistics also makes use of matrices in many different forms. Descriptive statisticsDescriptive statisticsDescriptive statistics quantitatively describe the main features of a collection of data. Descriptive statistics are distinguished from inferential statistics , in that descriptive statistics aim to summarize a data set, rather than use the data to learn about the population that the data are...
is concerned with describing data sets, which can often be represented in matrix form, by reducing the amount of data. The covariance matrixCovariance matrixIn probability theory and statistics, a covariance matrix is a matrix whose element in the i, j position is the covariance between the i th and j th elements of a random vector...
encodes the mutual varianceVarianceIn probability theory and statistics, the variance is a measure of how far a set of numbers is spread out. It is one of several descriptors of a probability distribution, describing how far the numbers lie from the mean . In particular, the variance is one of the moments of a distribution...
of several random variableRandom variableIn probability and statistics, a random variable or stochastic variable is, roughly speaking, a variable whose value results from a measurement on some type of random process. Formally, it is a function from a probability space, typically to the real numbers, which is measurable functionmeasurable...
s. Another technique using matrices are linear least squaresLinear least squaresIn statistics and mathematics, linear least squares is an approach to fitting a mathematical or statistical model to data in cases where the idealized value provided by the model for any data point is expressed linearly in terms of the unknown parameters of the model...
, a method that approximates a finite set of pairs (x_{1}, y_{1}), (x_{2}, y_{2}), ..., (x_{N}, y_{N}), by a linear function y_{i} ≈ ax_{i} + b, i = 1, ..., N
which can be formulated in terms of matrices, related to the singular value decompositionSingular value decompositionIn linear algebra, the singular value decomposition is a factorization of a real or complex matrix, with many useful applications in signal processing and statistics....
of matrices.
Random matricesRandom matrixIn probability theory and mathematical physics, a random matrix is a matrixvalued random variable. Many important properties of physical systems can be represented mathematically as matrix problems...
are matrices whose entries are random numbers, subject to suitable probability distributionProbability distributionIn probability theory, a probability mass, probability density, or probability distribution is a function that describes the probability of a random variable taking certain values....
s, such as matrix normal distributionMatrix normal distributionThe matrix normal distribution is a probability distribution that is a generalization of the normal distribution to matrixvalued random variables. Definition :...
. Beyond probability theory, they are applied in domains ranging from number theoryNumber theoryNumber theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
to physicsPhysicsPhysics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...
.
Symmetries and transformations in physics
Linear transformations and the associated symmetriesSymmetrySymmetry generally conveys two primary meanings. The first is an imprecise sense of harmonious or aesthetically pleasing proportionality and balance; such that it reflects beauty or perfection...
play a key role in modern physics. For example, elementary particleElementary particleIn particle physics, an elementary particle or fundamental particle is a particle not known to have substructure; that is, it is not known to be made up of smaller particles. If an elementary particle truly has no substructure, then it is one of the basic building blocks of the universe from which...
s in quantum field theoryQuantum field theoryQuantum field theory provides a theoretical framework for constructing quantum mechanical models of systems classically parametrized by an infinite number of dynamical degrees of freedom, that is, fields and manybody systems. It is the natural and quantitative language of particle physics and...
are classified as representations of the Lorentz groupLorentz groupIn physics , the Lorentz group is the group of all Lorentz transformations of Minkowski spacetime, the classical setting for all physical phenomena...
of special relativity and, more specifically, by their behavior under the spin group. Concrete representations involving the Pauli matricesPauli matricesThe Pauli matrices are a set of three 2 × 2 complex matrices which are Hermitian and unitary. Usually indicated by the Greek letter "sigma" , they are occasionally denoted with a "tau" when used in connection with isospin symmetries...
and more general gamma matrices are an integral part of the physical description of fermionFermionIn particle physics, a fermion is any particle which obeys the Fermi–Dirac statistics . Fermions contrast with bosons which obey Bose–Einstein statistics....
s, which behave as spinorSpinorIn mathematics and physics, in particular in the theory of the orthogonal groups , spinors are elements of a complex vector space introduced to expand the notion of spatial vector. Unlike tensors, the space of spinors cannot be built up in a unique and natural way from spatial vectors...
s. For the three lightest quarkQuarkA quark is an elementary particle and a fundamental constituent of matter. Quarks combine to form composite particles called hadrons, the most stable of which are protons and neutrons, the components of atomic nuclei. Due to a phenomenon known as color confinement, quarks are never directly...
s, there is a grouptheoretical representation involving the special unitary groupSpecial unitary groupThe special unitary group of degree n, denoted SU, is the group of n×n unitary matrices with determinant 1. The group operation is that of matrix multiplication...
SU(3); for their calculations, physicists use a convenient matrix representation known as the GellMann matricesGellMann matricesThe GellMann matrices, named for Murray GellMann, are one possible representation of the infinitesimal generators of the special unitary group called SU....
, which are also used for the SU(3) gauge group that forms the basis of the modern description of strong nuclear interactions, quantum chromodynamicsQuantum chromodynamicsIn theoretical physics, quantum chromodynamics is a theory of the strong interaction , a fundamental force describing the interactions of the quarks and gluons making up hadrons . It is the study of the SU Yang–Mills theory of colorcharged fermions...
. The Cabibbo–Kobayashi–Maskawa matrix, in turn, expresses the fact that the basic quark states that are important for weak interactionWeak interactionWeak interaction , is one of the four fundamental forces of nature, alongside the strong nuclear force, electromagnetism, and gravity. It is responsible for the radioactive decay of subatomic particles and initiates the process known as hydrogen fusion in stars...
s are not the same as, but linearly related to the basic quark states that define particles with specific and distinct massMassMass can be defined as a quantitive measure of the resistance an object has to change in its velocity.In physics, mass commonly refers to any of the following three properties of matter, which have been shown experimentally to be equivalent:...
es.
Linear combinations of quantum states
The first model of quantum mechanicsQuantum mechanicsQuantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particlelike and wavelike behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...
(HeisenbergWerner HeisenbergWerner Karl Heisenberg was a German theoretical physicist who made foundational contributions to quantum mechanics and is best known for asserting the uncertainty principle of quantum theory...
, 1925) represented the theory's operators by infinitedimensional matrices acting on quantum states. This is also referred to as matrix mechanicsMatrix mechanicsMatrix mechanics is a formulation of quantum mechanics created by Werner Heisenberg, Max Born, and Pascual Jordan in 1925.Matrix mechanics was the first conceptually autonomous and logically consistent formulation of quantum mechanics. It extended the Bohr Model by describing how the quantum jumps...
. One particular example is the density matrixDensity matrixIn quantum mechanics, a density matrix is a selfadjoint positivesemidefinite matrix of trace one, that describes the statistical state of a quantum system...
that characterizes the "mixed" state of a quantum system as a linear combination of elementary, "pure" eigenstates.
Another matrix serves as a key tool for describing the scattering experiments that form the cornerstone of experimental particle physics: Collision reactions such as occur in particle acceleratorParticle acceleratorA particle accelerator is a device that uses electromagnetic fields to propel charged particles to high speeds and to contain them in welldefined beams. An ordinary CRT television set is a simple form of accelerator. There are two basic types: electrostatic and oscillating field accelerators.In...
s, where noninteracting particles head towards each other and collide in a small interaction zone, with a new set of noninteracting particles as the result, can be described as the scalar product of outgoing particle states and a linear combination of ingoing particle states. The linear combination is given by a matrix known as the Smatrix, which encodes all information about the possible interactions between particles.
Normal modes
A general application of matrices in physics is to the description of linearly coupled harmonic systems. The equations of motionEquation of motionEquations of motion are equations that describe the behavior of a system in terms of its motion as a function of time...
of such systems can be described in matrix form, with a mass matrix multiplying a generalized velocity to give the kinetic term, and a force matrix multiplying a displacement vector to characterize the interactions. The best way to obtain solutions is to determine the system's eigenvectors, its normal modeNormal modeA normal mode of an oscillating system is a pattern of motion in which all parts of the system move sinusoidally with the same frequency and with a fixed phase relation. The frequencies of the normal modes of a system are known as its natural frequencies or resonant frequencies...
s, by diagonalizing the matrix equation. Techniques like this are crucial when it comes to the internal dynamics of molecules: the internal vibrations of systems consisting of mutually bound component atoms. They are also needed for describing mechanical vibrations, and oscillations in electrical circuits.
Geometrical optics
Geometrical opticsGeometrical opticsGeometrical optics, or ray optics, describes light propagation in terms of "rays". The "ray" in geometric optics is an abstraction, or "instrument", which can be used to approximately model how light will propagate. Light rays are defined to propagate in a rectilinear path as far as they travel in...
provides further matrix applications. In this approximative theory, the wave nature of light is neglected. The result is a model in which light rays are indeed geometrical rays. If the deflection of light rays by optical elements is small, the action of a lensLens (optics)A lens is an optical device with perfect or approximate axial symmetry which transmits and refracts light, converging or diverging the beam. A simple lens consists of a single optical element...
or reflective element on a given light ray can be expressed as multiplication of a twocomponent vector with a twobytwo matrix called ray transfer matrix: the vector's components are the light ray's slope and its distance from the optical axis, while the matrix encodes the properties of the optical element. Actually, there are two kinds of matrices, viz. a refraction matrix describing the refraction at a lens surface, and a translation matrix, describing the translation of the plane of reference to the next refracting surface, where another refraction matrix applies.
The optical system, consisting of a combination of lenses and/or reflective elements, is simply described by the matrix resulting from the product of the components' matrices.
Electronics
Traditional mesh analysisMesh analysisMesh analysis is a method that is used to solve planar circuits for the currents at any place in the circuit. Planar circuits are circuits that can be drawn on a plane surface with no wires crossing each other. A more general technique, called loop analysis can be applied to any circuit, planar...
in electronics leads to a system of linear equations that can be described with a matrix.
The behaviour of many electronicElectronicsElectronics is the branch of science, engineering and technology that deals with electrical circuits involving active electrical components such as vacuum tubes, transistors, diodes and integrated circuits, and associated passive interconnection technologies...
components can be described using matrices. Let A be a 2dimensional vector with the component's input voltage v_{1} and input current i_{1} as its elements, and let B be a 2dimensional vector with the component's output voltage v_{2} and output current i_{2} as its elements. Then the behaviour of the electronic component can be described by B = H · A, where H is a 2 x 2 matrix containing one impedanceElectrical impedanceElectrical impedance, or simply impedance, is the measure of the opposition that an electrical circuit presents to the passage of a current when a voltage is applied. In quantitative terms, it is the complex ratio of the voltage to the current in an alternating current circuit...
element (h_{12}), one admittanceAdmittanceIn electrical engineering, the admittance is a measure of how easily a circuit or device will allow a current to flow. It is defined as the inverse of the impedance . The SI unit of admittance is the siemens...
element (h_{21}) and two dimensionlessDimensionless quantityIn dimensional analysis, a dimensionless quantity or quantity of dimension one is a quantity without an associated physical dimension. It is thus a "pure" number, and as such always has a dimension of 1. Dimensionless quantities are widely used in mathematics, physics, engineering, economics, and...
elements (h_{11} and h_{22}). Calculating a circuit now reduces to multiplying matrices.
History
Matrices have a long history of application in solving linear equations. The Chinese textChinese mathematicsMathematics in China emerged independently by the 11th century BC. The Chinese independently developed very large and negative numbers, decimals, a place value decimal system, a binary system, algebra, geometry, and trigonometry....
The Nine Chapters on the Mathematical ArtThe Nine Chapters on the Mathematical ArtThe Nine Chapters on the Mathematical Art is a Chinese mathematics book, composed by several generations of scholars from the 10th–2nd century BCE, its latest stage being from the 1st century CE...
(Jiu Zhang Suan Shu), from between 300 BC and AD 200, is the first example of the use of matrix methods to solve simultaneous equations, including the concept of determinantDeterminantIn linear algebra, the determinant is a value associated with a square matrix. It can be computed from the entries of the matrix by a specific arithmetic expression, while other ways to determine its value exist as well...
s, over 1000 years before its publication by the Japanese mathematicianJapanese mathematicsdenotes a distinct kind of mathematics which was developed in Japan during the Edo Period . The term wasan, from wa and san , was coined in the 1870s and employed to distinguish native Japanese mathematics theory from Western mathematics .In the history of mathematics, the development of wasan...
Seki in 1683 and the German mathematician LeibnizGottfried LeibnizGottfried Wilhelm Leibniz was a German philosopher and mathematician. He wrote in different languages, primarily in Latin , French and German ....
in 1693. CramerGabriel CramerGabriel Cramer was a Swiss mathematician, born in Geneva. He showed promise in mathematics from an early age. At 18 he received his doctorate and at 20 he was cochair of mathematics.In 1728 he proposed a solution to the St...
presented his ruleCramer's ruleIn linear algebra, Cramer's rule is a theorem, which gives an expression for the solution of a system of linear equations with as many equations as unknowns, valid in those cases where there is a unique solution...
in 1750.
Early matrix theory emphasized determinants more strongly than matrices and an independent matrix concept akin to the modern notion emerged only in 1858, with Cayley'sArthur CayleyArthur Cayley F.R.S. was a British mathematician. He helped found the modern British school of pure mathematics....
Memoir on the theory of matrices. The term "matrix" (LatinLatinLatin is an Italic language originally spoken in Latium and Ancient Rome. It, along with most European languages, is a descendant of the ancient ProtoIndoEuropean language. Although it is considered a dead language, a number of scholars and members of the Christian clergy speak it fluently, and...
for "womb", derived from mater—mother) was coined by Sylvester, who understood a matrix as an object giving rise to a number of determinants today called minorMinor (linear algebra)In linear algebra, a minor of a matrix A is the determinant of some smaller square matrix, cut down from A by removing one or more of its rows or columns...
s, that is to say, determinants of smaller matrices that derive from the original one by removing columns and rows. In a 1851 paper, Sylvester explains: I have in previous papers defined a "Matrix" as a rectangular array of terms, out of which different systems of determinants may be engendered as from the womb of a common parent.
The study of determinants sprang from several sources. NumbertheoreticalNumber theoryNumber theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
problems led Gauss to relate coefficients of quadratic formQuadratic formIn mathematics, a quadratic form is a homogeneous polynomial of degree two in a number of variables. For example,4x^2 + 2xy  3y^2\,\!is a quadratic form in the variables x and y....
s, i.e., expressions such as and linear maps in three dimensions to matrices. Eisenstein further developed these notions, including the remark that, in modern parlance, matrix products are noncommutative. Cauchy was the first to prove general statements about determinants, using as definition of the determinant of a matrix A = [a_{i,j}] the following: replace the powers a_{j}^{k} by a_{jk} in the polynomialPolynomialIn mathematics, a polynomial is an expression of finite length constructed from variables and constants, using only the operations of addition, subtraction, multiplication, and nonnegative integer exponents...
where Π denotes the productMultiplicationMultiplication is the mathematical operation of scaling one number by another. It is one of the four basic operations in elementary arithmetic ....
of the indicated terms. He also showed, in 1829, that the eigenvalues of symmetric matrices are real. JacobiCarl Gustav Jakob JacobiCarl Gustav Jacob Jacobi was a German mathematician, widely considered to be the most inspiring teacher of his time and is considered one of the greatest mathematicians of his generation.Biography:...
studied "functional determinants"—later called Jacobi determinants by Sylvester—which can be used to describe geometric transformations at a local (or infinitesimalInfinitesimalInfinitesimals have been used to express the idea of objects so small that there is no way to see them or to measure them. The word infinitesimal comes from a 17th century Modern Latin coinage infinitesimus, which originally referred to the "infiniteth" item in a series.In common speech, an...
) level, see above; Kronecker'sLeopold KroneckerLeopold Kronecker was a German mathematician who worked on number theory and algebra.He criticized Cantor's work on set theory, and was quoted by as having said, "God made integers; all else is the work of man"...
Vorlesungen über die Theorie der Determinanten and Weierstrass'Karl WeierstrassKarl Theodor Wilhelm Weierstrass was a German mathematician who is often cited as the "father of modern analysis". Biography :Weierstrass was born in Ostenfelde, part of Ennigerloh, Province of Westphalia....
Zur Determinantentheorie, both published in 1903, first treated determinants axiomAxiomIn traditional logic, an axiom or postulate is a proposition that is not proven or demonstrated but considered either to be selfevident or to define and delimit the realm of analysis. In other words, an axiom is a logical statement that is assumed to be true...
atically, as opposed to previous more concrete approaches such as the mentioned formula of Cauchy. At that point, determinants were firmly established.
Many theorems were first established for small matrices only, for example the Cayley–Hamilton theoremCayley–Hamilton theoremIn linear algebra, the Cayley–Hamilton theorem states that every square matrix over a commutative ring satisfies its own characteristic equation....
was proved for 2×2 matrices by Cayley in the aforementioned memoir, and by HamiltonWilliam Rowan HamiltonSir William Rowan Hamilton was an Irish physicist, astronomer, and mathematician, who made important contributions to classical mechanics, optics, and algebra. His studies of mechanical and optical systems led him to discover new mathematical concepts and techniques...
for 4×4 matrices. Frobenius, working on bilinear forms, generalized the theorem to all dimensions (1898). Also at the end of the 19th century the Gauss–Jordan eliminationGauss–Jordan eliminationIn linear algebra, Gauss–Jordan elimination is an algorithm for getting matrices in reduced row echelon form using elementary row operations. It is a variation of Gaussian elimination. Gaussian elimination places zeros below each pivot in the matrix, starting with the top row and working downwards....
(generalizing a special case now known as Gauss elimination) was established by Jordan. In the early 20th century, matrices attained a central role in linear algebra. partially due to their use in classification of the hypercomplex numberHypercomplex numberIn mathematics, a hypercomplex number is a traditional term for an element of an algebra over a field where the field is the real numbers or the complex numbers. In the nineteenth century number systems called quaternions, tessarines, coquaternions, biquaternions, and octonions became established...
systems of the previous century.
The inception of matrix mechanicsMatrix mechanicsMatrix mechanics is a formulation of quantum mechanics created by Werner Heisenberg, Max Born, and Pascual Jordan in 1925.Matrix mechanics was the first conceptually autonomous and logically consistent formulation of quantum mechanics. It extended the Bohr Model by describing how the quantum jumps...
by HeisenbergWerner HeisenbergWerner Karl Heisenberg was a German theoretical physicist who made foundational contributions to quantum mechanics and is best known for asserting the uncertainty principle of quantum theory...
, BornMax BornMax Born was a Germanborn physicist and mathematician who was instrumental in the development of quantum mechanics. He also made contributions to solidstate physics and optics and supervised the work of a number of notable physicists in the 1920s and 30s...
and JordanPascual JordanFurther reading:...
led to studying matrices with infinitely many rows and columns. Later, von NeumannJohn von NeumannJohn von Neumann was a HungarianAmerican mathematician and polymath who made major contributions to a vast number of fields, including set theory, functional analysis, quantum mechanics, ergodic theory, geometry, fluid dynamics, economics and game theory, computer science, numerical analysis,...
carried out the mathematical formulation of quantum mechanicsMathematical formulation of quantum mechanicsThe mathematical formulations of quantum mechanics are those mathematical formalisms that permit a rigorous description of quantum mechanics. Such are distinguished from mathematical formalisms for theories developed prior to the early 1900s by the use of abstract mathematical structures, such as...
, by further developing functional analyticFunctional analysisFunctional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limitrelated structure and the linear operators acting upon these spaces and respecting these structures in a suitable sense...
notions such as linear operators on Hilbert spaceHilbert spaceThe mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the twodimensional Euclidean plane and threedimensional space to spaces with any finite or infinite number of dimensions...
s, which, very roughly speaking, correspond to Euclidean spaceEuclidean spaceIn mathematics, Euclidean space is the Euclidean plane and threedimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...
, but with an infinity of independent directions.
Other historical usages of the word “matrix” in mathematics
The word has been used in unusual ways by at least two authors of historical importance.
Bertrand RussellBertrand RussellBertrand Arthur William Russell, 3rd Earl Russell, OM, FRS was a British philosopher, logician, mathematician, historian, and social critic. At various points in his life he considered himself a liberal, a socialist, and a pacifist, but he also admitted that he had never been any of these things...
and Alfred North WhiteheadAlfred North WhiteheadAlfred North Whitehead, OM FRS was an English mathematician who became a philosopher. He wrote on algebra, logic, foundations of mathematics, philosophy of science, physics, metaphysics, and education...
in their Principia Mathematica (1910–1913) use the word matrix in the context of their Axiom of reducibilityAxiom of reducibilityThe axiom of reducibility was introduced by Bertrand Russell as part of his ramified theory of types, an attempt to ground mathematics in firstorder logic. History: the problem of impredicativity :...
. They proposed this axiom as a means to reduce any function to one of lower type, successively, so that at the “bottom” (0 order) the function is identical to its extensionExtension (predicate logic)The extension of a predicatea truthvalued functionis the set of tuples of values that, used as arguments, satisfy the predicate. Such a set of tuples is a relation.For example the statement "d2 is the weekday following d1"...
: “Let us give the name of matrix to any function, of however many variables, which does not involve any apparent variables. Then any possible function other than a matrix is derived from a matrix by means of generalization, i.e., by considering the proposition which asserts that the function in question is true with all possible values or with some value of one of the arguments, the other argument or arguments remaining undetermined”.
For example a function Φ(x, y) of two variables x and y can be reduced to a collection of functions of a single variable, e.g., y, by “considering” the function for all possible values of “individuals” a_{i} substituted in place of variable x. And then the resulting collection of functions of the single variable y, i.e., ∀a_{i}: Φ(a_{i}, y), can be reduced to a “matrix” of values by “considering” the function for all possible values of “individuals” b_{i} substituted in place of variable y: ∀b_{j}∀a_{i}: Φ(a_{i}, b_{j}).
Alfred TarskiAlfred TarskiAlfred Tarski was a Polish logician and mathematician. Educated at the University of Warsaw and a member of the LwowWarsaw School of Logic and the Warsaw School of Mathematics and philosophy, he emigrated to the USA in 1939, and taught and carried out research in mathematics at the University of...
in his 1946 Introduction to Logic used the word “matrix” synonymously with the notion of truth tableTruth tableA truth table is a mathematical table used in logic—specifically in connection with Boolean algebra, boolean functions, and propositional calculus—to compute the functional values of logical expressions on each of their functional arguments, that is, on each combination of values taken by their...
as used in mathematical logic.
See also
 Algebraic multiplicity
 Geometric multiplicity
 GramSchmidt process
 List of matrices
 Matrix calculusMatrix calculusIn mathematics, matrix calculus is a specialized notation for doing multivariable calculus, especially over spaces of matrices, where it defines the matrix derivative. This notation was to describe systems of differential equations, and taking derivatives of matrixvalued functions with respect...
 Periodic matrix setPeriodic matrix setIn mathematics, a periodic matrix set is a set of square matrices in which each square matrix is of a different size, and such that each cell within each matrix within a set contains data associated with some type of periodic distribution....
 TensorTensorTensors are geometric objects that describe linear relations between vectors, scalars, and other tensors. Elementary examples include the dot product, the cross product, and linear maps. Vectors and scalars themselves are also tensors. A tensor can be represented as a multidimensional array of...
External links
History MacTutor: Matrices and determinants
 Matrices and Linear Algebra on the Earliest Uses Pages
 Earliest Uses of Symbols for Matrices and Vectors
Online books
Online matrix calculators
, a freeware package for matrix algebra and statistics
