I know some basic property of determinant. I read an article and see this formula: \begin{equation} |(\delta-p-1)D+S|=\bigg(\prod_{i=1}^{p}\lambda_{i}(D)\bigg) \bigg(\prod_{i=1}^{p}(\delta-p-1)\lambda_{i}(D^{-1}S)\bigg) \end{equation} $D$ and $S$ is $p\times p$ and a symmetric matrix, $\lambda_{i}(M)$ represents $i$th largest eigenvalues of $M$, and $\delta>p+1$. When I implement above formula in R the above formula is not established.
Can anyone give hints?
Thank you
It looks like $D$ can be assumed to be invertible so we have $$ (\delta - p -1) D + S = D\left((\delta - p - 1) I + D^{-1}S\right) $$ and by the fact that $|AB|=|A||B|$ we have $$ |(\delta - p -1) D + S| = |D||(\delta - p - 1) I + D^{-1}S| \\ = \left(\prod_{i=1}^p \lambda_i(D)\right)\left(\prod_{i=1}^p (\delta - p - 1) + \lambda_i(D^{-1}S)\right) $$ so maybe there is a typo in the formula and it should have $\delta - p - 1$ being added instead of multiplied?
