# Taylor expansion for matrix logarithm

1. Mar 29, 2013

### Backpacker

A paper I'm reading states the that: for positive hermitian matrices A and B, the Taylor expansion of $\log(A+tB)$ at t=0 is

$\log(A+tB)=\log(A) + t\int_0^\infty \frac{1}{B+zI}A \frac{1}{B+zI} dz + \mathcal{O}(t^2).$

However, there is no source or proof given, and I cannot seem to find a derivation of this identity anywhere! Any help would be appreciated. Thanks.

Last edited: Mar 29, 2013
2. Mar 29, 2013

### I like Serena

Welcome to PF, Backpacker!

I don't recognize your formula, but:

$$\log(A+tB)=\log(A(I+tA^{-1}B)= \log A + \log(I+tA^{-1}B) = \log A + tA^{-1}B + \mathcal{O}(t^2)$$

3. Sep 11, 2015

This doesn't seem quite right, unless $A$ and $B$ commute.