I've got an expression of the form
\begin{equation}\det(I-AB)\end{equation} and I'm wondering if there is a way to write this solely in terms of functions of $A$ and $B$. For the particular case I'm considering, $A$ is some diagonal matrix and $B$ is some symmetric matrix with zeroes on the diagonal, so I would think since the matrix $I-AB$ only differs by some diagonal elements and a few other things it would be possible to construct this determinant from $A$ and $B$ in general. Also the inverse of such a matrix, i.e. $(I-AB)^{-1}$ would be useful to know as well. In general, $||AB||>1$ so I don't think I can expand the inverse in powers of $AB$ the usual way.
Anyways, thanks for any help.
[Math] Determinant and Inverse of a Difference of two matrices
linear algebramatrices
Best Answer
The product $A \cdot B$ is a symmetric matrix again. If I summarize your question: Given a symmetric matrix $B$ with zeros on the diagonal, is there a simple way to compute $\det(I - B)$.
Answer: There is none in general. A good approximation for a $B$ with only small entries is $$\det(I - B) = 1 - {\rm trace} (B).$$
However, consider that $I - B$ has only $1$'s on the diagonal. Thus, it should be relatively easy to reduce the problem to lower dimensions.