A _matrix logarithm_ is a [matrix function](Functions%20of%20linear%20operators.md#matrix%20functions) that is the inverse of a [matrix exponential](Matrix%20exponentials.md) and thus for a [function of an operator, $X,$](Functions%20of%20linear%20operators.md) it is expressed as $f(X)=\ln X.$ # Power series of matrix logarithms Given an operator $X$ or a matrix $X$ with real or complex elements, in order to evaluate its [logarithm](Matrix%20logarithms.md), we need to first write it as a [power series](Functions%20of%20linear%20operators.md#as%20a%20power%20series), $\ln X = \sum_{n=1}^{\infty} (-1)^{n+1} \frac{(X - I)^n}{n},$ which follows from the expression of any function, $\ln x$ [as a power series](Analysis%20(index).md#The%20natural%20log%20as%20an%20infinite%20series). Here also we need to, as with the case where we take the $\ln$ of a function include the condition for which the series converges. With matrices this means taking the [matrix norm](Operator%20norm.md) $\| X - I \| < 1.$ Note that this is the condition for which the power series _always_ converges. It may also converge [when $\| X - I \| > 1.$ ](Matrix%20logarithms.md#Cases%20where%20the%20[power%20series%20of%20matrix%20logarithms](Matrix%20logarithms.md%20Power%20series%20of%20matrix%20logarithms)%20converges%20for%20$%20X%20-%20I%20>%201.$) ## Cases where the [power series of matrix logarithms](Matrix%20logarithms.md#Power%20series%20of%20matrix%20logarithms) converges for $\| X - I \| > 1.$ # Properties of matrix logarithms --- # Proofs and Examples ## Matrix logarithm of a nilpotent matrix #MathematicalFoundations/Algebra/AbstractAlgebra/LinearAlgebra/Operators/Matrices #MathematicalFoundations/Algebra/AbstractAlgebra/GroupTheory/Lie/LieGroups