I have an $ n \times n$ matrix B that I need to take the determinant of, but B is badly scaled so that numerically (via Matlab) its determinant is -Inf. However, B is constructed by collapsing an $ m \times m$ matrix H with an $n \times m$ matrix A, $$ B = A H A'%$$ where $m > n$.
My question is if there is a way to find the determinant of B without computing it explicitly. I have already computed the determinant of H. I'm imagining that there's some simple decomposition to get the determinant of B given A and the determinant of H but I can't find anything on this.
A few (potentially helpful) notes:
- H is symmetric and positive definite (and as a consequence of construction, so is B).
- At the end of the day I really care about the log ratio of the determinants, $\log \frac{|H|}{|B|}$, but I'm assuming that the easiest way to do this is $\log |H| - \log |B|$. The only reason I have the determinant of H mentioned above is for this purpose, so kudos if you have an efficient way of doing so.