matrix identity for $\log(\det(1+AB))$ where B is symmetric and A is diagonal

Question

I'm wondering if there are matrix identities that make it easy to compute things like $\log(\det(I+AB))$ where $B$ is a symmetric matrix, and $A$ is a diagonal matrix. I know you can rewrite this as: $Tr(\log(I+AB))=AB -\frac{(AB)^2}{2}...$ but I'm curious if there are any simpler identities.

Answer 1

The power series for $\log(I+C)$ converges if $C$ has spectral radius $< 1$, but not if it has spectral radius $>1$.

If the eigenvalues of $AB$ are $\lambda_1, \ldots, \lambda_n$ (counted by algebraic multiplicity), $\log(\det(I+AB)) = \sum_{j=1}^n \log(1+\lambda_j) + 2 \pi i m$ for appropriate integer $m$ (depending on which branch of $\log$ you use...).