# Obscure determinant question

1. Aug 14, 2009

### grawil

I'm working on some math that falls out of using a Kalman filter for estimation. I'd like to show that $$|G^T C_1^{-1} G + C_2^{-1}| \leq |G^T [diag(C_1)]^{-1} G + C_2^{-1}|$$ where $$C_1$$ and $$C_2$$ are covariance matrices and $$diag(C_1)$$ denotes the diagonal of matrix A.

I've been able to show this is true when $$G=G^T=I$$ and $$C_2$$ is also diagonal. Numerical simulations suggest it is more generally true but I've been unable to convince myself of it on paper.