$\log (A + \delta A) = ?$ (as an expansion in $\delta A$), where $A$ and $\delta A $ are matrices

Question

$A$ and $\delta A$ are two non-commuting matrices and I am seeking a power series expansion to 2nd order in $\delta A$.

After writing it as $\log (A (1 + A^{-1}\delta A) )$, I am unable to figure out how to simplify further, since, the matrices don't commute.

Any hints or suggestions will be greatly useful.

Answer 1

Writing $-\log(A) = \int d \beta \left[ \frac{1}{A+ \beta} - \frac{1}{1+\beta} \right] $, helps. By substituting $A \to A+\delta A$, and using binomial expansion, one gets an integral representation for the 2nd order term, which is what I was looking for.

Thanks!

Answer 2

@ Srivatsan Balakrishnan , you write anything.

We assume that $A$ has no eigenvalues in $(-\infty,0)]$ and $\log(.)$ is the principal logarithm. The Taylor formula is: $\log(A+H)=\log(A)+Df_A(H)+\dfrac{1}{2}D^2f_A(H,H)+o(||H||^2)$.

Now we calculate the first and second derivatives of the matrix function $f(A)=\log(A)$:

$Df_A(H)=\int_0^1(t(A-I)+I)^{-1}H(t(A-I)+I)^{-1}dt$.

$D^2f_A(H,K)=-\int_0^1(t(t(A-I)+I)^{-1}K(t(A-I)+I)^{-1}H(t(A-I)+I)^{-1}+t(t(A-I)+I)^{-1}H(t(A-I)+I)^{-1}K(t(A-I)+I)^{-1})dt$

$D^2f_A(H,H)=-2\int_0^1t(t(A-I)+I)^{-1}H(t(A-I)+I)^{-1}H(t(A-I)+I)^{-1}dt$