I'm working on a comparison between two algorithms that I have developed. It seems like the one gets better results than the other, and I'm trying to give this observation a theoretical guarantee, if possible.
I'm trying to obtain the following result:
Consider a real matrix $P \succ0$ which is positive definite. We denote by $D$ the diagonal matrix with entries $P_{11}^{-1},\ldots,P_{nn}^{-1}$ on the diagonal.
I suspect that the matrix $DPD-2D+P^{-1}$ is PSD.
I tried to use several notions, but with no results. I also did not manage to disprove this fact. Any ideas?
Yes, it is always positive semidefinite. Let $S=D^{1/2}PD^{1/2}$. Then $S\succ0$ and \begin{align} DPD - 2D + P^{-1} &=D^{1/2}\left(S - 2I + S^{-1}\right)D^{1/2}\\ &=D^{1/2}\left(S^{1/2} - S^{-1/2}\right)^2D^{1/2}\tag{1}\\ &\succeq0. \end{align} The expression in $(1)$ shows that $DPD - 2D + P^{-1}\succ0$ if and only if $1$ is not an eigenvalue of $S$.