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.