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 logarithmMathworldPlanetmathPlanetmath. 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 seriesMathworldPlanetmath when A is a square matrixMathworldPlanetmath and ||I-A||F<1, where ||||F is the Frobenius matrix norm. The logarithm this formula produces is known as the principal logarithm of A.


Since this series expansion does not converge for all A, it is not a global inverse function for the matrix exponentialMathworldPlanetmath. In particular, explogA=A only holds for ||I-A||F<1, and log(expA)=A only holds for ||A||F<2.

There are other, more general methods of calculating the matrix logarithm. For example, see \htmladdnormallinkAn Explicit Formula for the Matrix Logarithm

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