Centrosymmetric matrix
Encyclopedia
In mathematics
, especially in linear algebra
and matrix theory, a centrosymmetric matrix is a matrix
which is symmetric about its center. More precisely, an n × n matrix A = [ Ai,j ] is centrosymmetric when its entries satisfy
If J denotes the n × n matrix with 1 on the counterdiagonal and 0 elsewhere (that is, Ji,n+1-i = 1; Ji,j = 0 if j ≠ n+1-i), then a matrix A is centrosymmetric if and only if AJ = JA. The matrix J is sometimes referred to as the exchange matrix
.
K, then so are A+B and cA for any c in K. In addition, the matrix product AB is centrosymmetric, since JAB = AJB = ABJ. Since the identity matrix
is also centrosymmetric, it follows that the set of n × n centrosymmetric matrices over K is a subalgebra of the associative algebra
of all n × n matrices.
The centrosymmetric relation AJ = JA lends itself to a natural generalization (see e.g.
)
, where J is replaced with an involutary matrix K (i.e., K2 = I).
Symmetric centrosymmetric matrices are sometimes called bisymmetric matrices. When the ground field
is the field of real numbers, it has been shown that bisymmetric matrices are precisely those symmetric matrices whose eigenvalues are the same up to sign after pre or post multiplication by the exchange matrix. A similar result holds for Hermitian centrosymmetric and skew-centrosymmetric matrices.
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...
, especially in linear algebra
Linear algebra
Linear algebra is a branch of mathematics that studies vector spaces, also called linear spaces, along with linear functions that input one vector and output another. Such functions are called linear maps and can be represented by matrices if a basis is given. Thus matrix theory is often...
and matrix theory, a centrosymmetric matrix is a matrix
Matrix (mathematics)
In mathematics, a matrix 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 isMatrices of the same size can be added or subtracted element by element...
which is symmetric about its center. More precisely, an n × n matrix A = [ Ai,j ] is centrosymmetric when its entries satisfy
- Ai,j = An−i+1,n−j+1 for 1 ≤ i,j ≤ n.
If J denotes the n × n matrix with 1 on the counterdiagonal and 0 elsewhere (that is, Ji,n+1-i = 1; Ji,j = 0 if j ≠ n+1-i), then a matrix A is centrosymmetric if and only if AJ = JA. The matrix J is sometimes referred to as the exchange matrix
Exchange matrix
In mathematics, especially linear algebra, the exchange matrix is a special case of a permutation matrix, where the 1 elements reside on the counterdiagonal and all other elements are zero...
.
Examples
- All 2×2 centrosymmetric matrices have the form
- All 3×3 centrosymmetric matrices have the form
- Symmetric ToeplitzToeplitz matrixIn linear algebra, a Toeplitz matrix or diagonal-constant matrix, named after Otto Toeplitz, is a matrix in which each descending diagonal from left to right is constant...
matrices are centrosymmetric.
Algebraic structure
If A and B are centrosymmetric matrices over a given 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...
K, then so are A+B and cA for any c in K. In addition, the matrix product AB is centrosymmetric, since JAB = AJB = ABJ. Since the identity matrix
Identity matrix
In 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...
is also centrosymmetric, it follows that the set of n × n centrosymmetric matrices over K is a subalgebra of the associative algebra
Associative algebra
In 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...
of all n × n matrices.
Related structures
An n × n matrix A is said to be skew-centrosymmetric if its entries satisfy Ai,j = -An−i+1,n−j+1 for 1 ≤ i,j ≤ n. Equivalently, A is skew-centrosymmetric if AJ = -JA, where J is the exchange matrix defined above.The centrosymmetric relation AJ = JA lends itself to a natural generalization (see e.g.
)
, where J is replaced with an involutary matrix K (i.e., K2 = I).
Symmetric centrosymmetric matrices are sometimes called bisymmetric matrices. When the ground field
Field (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...
is the field of real numbers, it has been shown that bisymmetric matrices are precisely those symmetric matrices whose eigenvalues are the same up to sign after pre or post multiplication by the exchange matrix. A similar result holds for Hermitian centrosymmetric and skew-centrosymmetric matrices.
External links
- Centrosymmetric matrix on MathWorldMathWorldMathWorld is an online mathematics reference work, created and largely written by Eric W. Weisstein. It is sponsored by and licensed to Wolfram Research, Inc. and was partially funded by the National Science Foundation's National Science Digital Library grant to the University of Illinois at...
.