Logarithm of a matrix - AbsoluteAstronomy.com

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 logarithm of a matrix is another 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...

such that the matrix exponential

Matrix exponential

In 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....

of the latter matrix equals the original matrix. It is thus a generalization of the scalar logarithm

Logarithm

The logarithm of a number is the exponent by which another fixed value, the base, has to be raised to produce that number. For example, the logarithm of 1000 to base 10 is 3, because 1000 is 10 to the power 3: More generally, if x = by, then y is the logarithm of x to base b, and is written...

and in some sense an inverse function

Inverse function

In mathematics, an inverse function is a function that undoes another function: If an input x into the function ƒ produces an output y, then putting y into the inverse function g produces the output x, and vice versa. i.e., ƒ=y, and g=x...

of the matrix exponential

Matrix exponential

. Not all matrices have a logarithm and those matrices that do have a logarithm may have more than one logarithm. The study of logarithms of matrices leads to Lie theory

Lie theory

Lie theory is an area of mathematics, developed initially by Sophus Lie.Early expressions of Lie theory are found in books composed by Lie with Friedrich Engel and Georg Scheffers from 1888 to 1896....

since when a matrix has a logarithm then it is in a Lie group

Lie group

In mathematics, a Lie group is a group which is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure...

and the logarithm is the corresponding element of the Lie algebra

Lie algebra

In mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds. Lie algebras were introduced to study the concept of infinitesimal transformations. The term "Lie algebra" was introduced by Hermann Weyl in the...

Definition

A matrix B is a logarithm of a given matrix A if the matrix exponential

Matrix exponential

of B is A:

e^B = A. \,

Example: Logarithm of rotations in the plane

The rotations in the plane give a simple example. A rotation of angle α around the origin is represented by the 2×2-matrix

A =
\begin{pmatrix}
\cos(\alpha) & -\sin(\alpha) \\
\sin(\alpha) & \cos(\alpha) \\
\end{pmatrix}.

For any integer n, the matrix

B_n=(\alpha+2\pi n)
\begin{pmatrix}
0 & -1 \\
1 & 0\\
\end{pmatrix},

is a logarithm of A. Thus, the matrix A has infinitely many logarithms. This corresponds to the fact that the rotation angle is only determined up to multiples of 2π. In the language of Lie theory, the rotation matrices A are elements of the Lie group SO(2). The corresponding logarithms B are elements of the Lie algebra so(2), which consists of all skew-symmetric matrices

Skew-symmetric matrix

In mathematics, and in particular linear algebra, a skew-symmetric 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...

. The matrix

\begin{pmatrix}
0 & 1 \\
-1 & 0\\
\end{pmatrix}

is a generator of the Lie algebra

Lie algebra

so(2).

Existence

The question of whether a matrix has a logarithm has the easiest answer when considered in the complex setting. A matrix has a logarithm if and only if

If and only if

In logic and related fields such as mathematics and philosophy, if and only if is a biconditional logical connective between statements....

it is invertible. The logarithm is not unique, but if a matrix has no negative real eigenvalues, then it has a unique logarithm whose eigenvalues lie all in the strip {z ∈ C | −π < Im z < π}. This logarithm is known as the principal logarithm. The answer is more involved in the real setting. A real matrix has a real logarithm if and only if it is invertible and each Jordan block belonging to a negative eigenvalue occurs an even number of times. If an invertible real matrix does not satisfy the condition with the Jordan blocks, then it has only complex logarithms. This can already be seen in the scalar case: the logarithm of −1 is a complex number. The existence of real matrix logarithms of real 2 x 2 matrices is considered in a later section.

Properties

If A and B are both positive-definite matrices and A and B commute, i.e., AB = BA, then

AB = e^{\ln(A)+\ln(B)}. \,

For any invertible matrix,

A^{-1} = e^{-\ln(A)}. \,

Further example: Logarithm of rotations in 3D space

A rotation

R\in {\rm SO(3)}

in the 3D is given by a 3x3 orthogonal matrix The logarithm of the rotation matrix R can be computed with Rodrigues' rotation formula

Rodrigues' rotation formula

In the theory of three-dimensional rotation, Rodrigues' rotation formula is an efficient algorithm for rotating a vector in space, given an axis and angle of rotation. By extension, this can be used to transform all three basis vectors to compute a rotation matrix from an axis-angle representation...

(see also Axis angle

Axis angle

The axis-angle representation of a rotation, also known as the exponential coordinates of a rotation, parameterizes a rotation by two values: a unit vector indicating the direction of a directed axis , and an angle describing the magnitude of the rotation about the axis...

\ln (R) = \left\{ \begin{matrix}
0 & \mathrm{if} \; \theta = 0 \\
\frac{\theta}{2 \sin(\theta)} (R - R^\top) & \mathrm{if} \; \theta \ne 0 \; \mathrm{and} \; \theta \in (-\pi, \pi)
  \end{matrix}\right.

Except when R has eigenvalues equal to −1 where the log is not unique. However, even in the case where

\theta = \pi

the Frobenius norm of the log is:

\| \ln(R) \|_F = \sqrt{2} | \theta |

Note that, given rotation matrices A and B:

d_g(A,B) := \| \log(A^\top B)\|_F

is the geodesic distance on the 3D manifold of rotation matrices.

Calculating the logarithm of a diagonalizable matrix

A method for finding ln A for a diagonalizable matrix

Diagonalizable matrix

In 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...

A is the following:NEWLINE

NEWLINE

Find the matrix V of eigenvectors of A (each column of V is an eigenvector of A).

NEWLINE

Find the inverse V⁻¹ of V.

NEWLINE

LetNEWLINE

NEWLINE

A' = V^{-1}  A  V.\,

NEWLINE

Then A′ will be a diagonal matrix whose diagonal elements are eigenvalues of A.

NEWLINE

Replace each diagonal element of A′ by its (natural) logarithm in order to obtain

\ln  A'

NEWLINE

ThenNEWLINE

NEWLINE

\ln A = V ( \ln A' ) V^{-1}. \,

NEWLINE

NEWLINE That the logarithm of A might be a complex matrix even if A is real then follows from the fact that a matrix with real and positive entries might nevertheless have negative or even complex eigenvalues (this is true for example for rotation matrices). The non-uniqueness of the logarithm of a matrix follows from the non-uniqueness of the logarithm of a complex number.

The logarithm of a non-diagonalizable matrix

The algorithm illustrated above does not work for non-diagonalizable matrices, such as

\begin{bmatrix}1 & 1\\ 0 & 1\end{bmatrix}.

For such matrices one needs to find its Jordan decomposition

Jordan normal form

In linear algebra, a Jordan normal form of a linear operator on a finite-dimensional vector space is an upper triangular matrix of a particular form called Jordan matrix, representing the operator on some basis...

and, rather than computing the logarithm of diagonal entries as above, one would calculate the logarithm of the Jordan block

Jordan matrix

In the mathematical discipline of matrix theory, a Jordan block over a ring R is a matrix composed of 0 elements everywhere except for the diagonal, which is filled with a fixed element \lambda\in R, and for the superdiagonal, which is composed of ones...

s. The latter is accomplished by noticing that one can write a Jordan block as

B=\begin{pmatrix}
\lambda & 1       & 0       & 0      & \cdots  & 0 \\
0       & \lambda & 1       & 0      & \cdots  & 0 \\
0       & 0       & \lambda & 1      & \cdots  & 0 \\
\vdots  & \vdots  & \vdots  & \ddots & \ddots  & \vdots \\
0       & 0       & 0       & 0      & \lambda & 1       \\
0       & 0       & 0       & 0      & 0       & \lambda \\\end{pmatrix}
\lambda \begin{pmatrix}
1 & \lambda^{-1}       & 0       & 0      & \cdots  & 0 \\
0       & 1 & \lambda^{-1}       & 0      & \cdots  & 0 \\
0       & 0       & 1 & \lambda^{-1}      & \cdots  & 0 \\
\vdots  & \vdots  & \vdots  & \ddots & \ddots  & \vdots \\
0       & 0       & 0       & 0      & 1 & \lambda^{-1}       \\
0       & 0       & 0       & 0      & 0       & 1 \\\end{pmatrix}=\lambda(I+K)

where K is a matrix with zeros on and under the main diagonal. (The number λ is nonzero by the assumption that the matrix whose logarithm one attempts to take is invertible.) Then, by the formula

\ln (1+x)=x-\frac{x^2}{2}+\frac{x^3}{3}-\frac{x^4}{4}+\cdots

one gets

\ln B=\ln \big(\lambda(I+K)\big)=\ln (\lambda I) +\ln (I+K)= (\ln \lambda) I + K-\frac{K^2}{2}+\frac{K^3}{3}-\frac{K^4}{4}+\cdots

This series

Series (mathematics)

A series is the sum of the terms of a sequence. Finite sequences and series have defined first and last terms, whereas infinite sequences and series continue indefinitely....

in general does not converge for every matrix K, as it would not for any real number with absolute value greater than unity, however, this particular K is a nilpotent

Nilpotent

In mathematics, an element x of a ring R is called nilpotent if there exists some positive integer n such that xn = 0....

matrix, so the series actually has a finite number of terms (K^m is zero if m is the dimension of K). Using this approach one finds

\ln \begin{bmatrix}1 & 1\\ 0 & 1\end{bmatrix}

A functional analysis perspective

A square matrix represents a linear operator on the Euclidean space

Euclidean space

In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...

Rⁿ where n is the dimension of the matrix. Since such a space is finite-dimensional, this operator is actually bounded

Bounded operator

In functional analysis, a branch of mathematics, a bounded linear operator is a linear transformation L between normed vector spaces X and Y for which the ratio of the norm of L to that of v is bounded by the same number, over all non-zero vectors v in X...

. Using the tools of holomorphic functional calculus

Holomorphic functional calculus

In mathematics, holomorphic functional calculus is functional calculus with holomorphic functions. That is to say, given a holomorphic function ƒ of a complex argument z and an operator T, the aim is to construct an operatorf\,...

, given a holomorphic function

Holomorphic function

In mathematics, holomorphic functions are the central objects of study in complex analysis. A holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighborhood of every point in its domain...

f(z) defined on an open set

Open set

The concept of an open set is fundamental to many areas of mathematics, especially point-set topology and metric topology. Intuitively speaking, a set U is open if any point x in U can be "moved" a small amount in any direction and still be in the set U...

in the complex plane

Complex plane

In mathematics, the complex plane or z-plane is a geometric representation of the complex numbers established by the real axis and the orthogonal imaginary axis...

and a bounded linear operator T, one can calculate f(T) as long as f(z) is defined on the spectrum of T. The function f(z)=ln z can be defined on any simply connected open set in the complex plane not containing the origin, and it is holomorphic on such a domain. This implies that one can define ln T as long as the spectrum of T does not contain the origin and there is a path going from the origin to infinity not crossing the spectrum of T (as such, if the spectrum of T is a circle with the origin inside of it, it is impossible to define ln T). Back to the particular case of a Euclidean space, the spectrum of a linear operator on this space is the set of eigenvalues of its matrix, and so is a finite set. As long as the origin is not in the spectrum (the matrix is invertible), one obviously satisfies the path condition from the previous paragraph, and as such, the theory implies that ln T is well-defined. The non-uniqueness of the matrix logarithm then follows from the fact that one can choose more than one branch of the logarithm which is defined on the set of eigenvalues of a matrix.

A Lie group theory perspective

In the theory of Lie group

Lie group

In mathematics, a Lie group is a group which is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure...

s, there is an exponential map

Exponential map

In differential geometry, the exponential map is a generalization of the ordinary exponential function of mathematical analysis to all differentiable manifolds with an affine connection....

from a Lie algebra

Lie algebra

g to the corresponding Lie group G NEWLINE

NEWLINE

\exp : g \rightarrow G.

NEWLINE For matrix Lie groups, the elements of g and G are square matrices and the exponential map is given by the matrix exponential

Matrix exponential

. The inverse map

\log=\exp^{-1}

is multivalued and coincides with the matrix logarithm discussed here. The logarithm maps from the Lie group G into the Lie algebra g. Note that the exponential map is a local diffeomorphism between a neighborhood U of the zero matrix

\underline{0} \in g

and a neighborhood V of the identity matrix

\underline{1}\in G

. Thus the (matrix) logarithm is well-defined as a map

\log: V\subset G\rightarrow U\subset g. \,

Constraints in 2 × 2 case

If a 2 x 2 real matrix has a negative 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...

, it has no real logarithm. Note first that any 2 × 2 real matrix can be considered one of the three types of complex number z = x + y ε where

\epsilon^2 = -1

, 0 , or + 1. This z is a point on a complex subplane of the ring

Ring (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...

of matrices. The case where the determinant is negative only arises in a plane with

\epsilon^2 = +1

, that is a split-complex number

Split-complex number

In abstract algebra, the split-complex numbers are a two-dimensional commutative algebra over the real numbers different from the complex numbers. Every split-complex number has the formwhere x and y are real numbers...

plane. Only one quarter of this plane is the image of the exponential map, so logarithm is only defined on that quarter (quadrant). The other three quadrants are images of this one under the Klein four-group

Klein four-group

In mathematics, the Klein four-group is the group Z2 × Z2, the direct product of two copies of the cyclic group of order 2...

generated by ε and −1. For example, let a = ln 2, then cosh a = 5/4 and sinh a = 3/4. For matrices this means

\exp \begin{pmatrix}0 & a \\ a & 0 \end{pmatrix} =
\begin{pmatrix}\cosh a & \sinh a \\ \sinh a & \cosh a  \end{pmatrix} =
\begin{pmatrix}1.25 & .75\\ .75 & 1.25 \end{pmatrix}

. So this last matrix has logarithm

\begin{pmatrix}0 & \ln 2 \\ \ln 2 & 0 \end{pmatrix}

. These matrices do not have a logarithm:

\begin{pmatrix}3/4 & 5/4 \\ 5/4 & 3/4 \end{pmatrix},\
\begin{pmatrix}-3/4 & -5/4 \\ -5/4 & -3/4\end{pmatrix}, \
\begin{pmatrix}-5/4 & -3/4\\ -3/4 & -5/4 \end{pmatrix}

. They represent three other conjugates by the four-group of the matrix above that does have a logarithm. A non-singular 2 x 2 matrix does not necessarily have a logarithm, but it is conjugate by the four-group to a matrix that does have a logarithm. For a richer affirmative example, start with a pythagorean triple

Pythagorean triple

A Pythagorean triple consists of three positive integers a, b, and c, such that . Such a triple is commonly written , and a well-known example is . If is a Pythagorean triple, then so is for any positive integer k. A primitive Pythagorean triple is one in which a, b and c are pairwise coprime...

(p,q,r) and let a = ln(p + r) − ln q. Then

e^a = \frac {p + r} {q} = \cosh a + \sinh a

. Now

\exp \begin{pmatrix}0 & a \\ a & 0 \end{pmatrix} =
\begin{pmatrix}r/q & p/q \\ p/q & r/q \end{pmatrix}

. Thus

\tfrac{1}{q}\begin{pmatrix}r & p \\ p & r \end{pmatrix}

has logarithm matrix

\begin{pmatrix}0 & a \\ a & 0 \end{pmatrix}

where

a = \ln(p+q) - \ln q

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