matrix logarithm

Unlike the scalar logarithm, there are no naturally-defined bases for the matrix logarithm; therefore, the matrix logarithm is always taken to be the natural logarithm. In general, there may be an infinite number of matrices $B$ satisfying $\exp (B)=A$; these are known as the logarithms of $A$.

As for the scalar natural logarithm, the matrix logarithm can be defined as a power series when $A$ is a square matrix and , where $||\cdot |{|}_F$ is the Frobenius matrix norm. The logarithm this formula produces is known as the principal logarithm of $A$.

$$\log (A)=-\sum _{k=1}^{\mathrm{\infty }}\frac{{(I-A)}^k}{k}=\log (I+X)=\sum _{k=1}^{\mathrm{\infty }}\frac{{(-1)}^{k+1}}{k}{X}^k$$

Since this series expansion does not converge for all $A$, it is not a global inverse function for the matrix exponential. In particular, ${\exp }^{\log A}=A$ only holds for , and $\log ({\exp }^A)=A$ only holds for .

There are other, more general methods of calculating the matrix logarithm. For example, see An Explicit Formula for the Matrix Logarithmhttp://arxiv.org/abs/math/0410556.

Title	matrix logarithm
Canonical name	MatrixLogarithm
Date of creation	2013-03-22 15:31:22
Last modified on	2013-03-22 15:31:22
Owner	Andrea Ambrosio (7332)
Last modified by	Andrea Ambrosio (7332)
Numerical id	11
Author	Andrea Ambrosio (7332)
Entry type	Definition
Classification	msc 15A90
Classification	msc 15A99
Related topic	NaturalLogarithm2
Related topic	MatrixFNorm
Related topic	FrobeniusMatrixNorm
Defines	principal logarithm