# 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 $||I-A||_{F}<1$, where $||\cdot||_{F}$ is the Frobenius matrix norm. The logarithm this formula produces is known as the principal logarithm of $A$.

 $\log(A)=-\sum_{k=1}^{\infty}\frac{(I-A)^{k}}{k}=\log(I+X)=\sum_{k=1}^{\infty}% \frac{(-1)^{k+1}}{k}X^{k}$

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 $||I-A||_{F}<1$, and $\log(\exp^{A})=A$ 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 Logarithmhttp://arxiv.org/abs/math/0410556.

Title matrix logarithm MatrixLogarithm 2013-03-22 15:31:22 2013-03-22 15:31:22 Andrea Ambrosio (7332) Andrea Ambrosio (7332) 11 Andrea Ambrosio (7332) Definition msc 15A90 msc 15A99 NaturalLogarithm2 MatrixFNorm FrobeniusMatrixNorm principal logarithm