Consider a symmetric positive definite matrix $P$ and arbitrary matrix $R<-I$. Does the following inequality hold? $$ P < RPR^T $$
If yes, provide some references. If no, guide me under what conditions on matrix $R$ the aforementioned inequality holds.
Thanks
It does not hold in general. For a counterexample, consider $$ R=R^T=-\pmatrix{4&1\\ 1&4},\quad D=\pmatrix{1\\ &16},\quad P=D^2,\quad x=\pmatrix{-4\\ 1}. $$ The spectrum of $R$ is $\{-5,-3\}$ (with eigenvectors $(1,\pm1)^T$), so that $R\prec -I$. However, we have $$ Dx=\pmatrix{-4\\ 16}\ \text{ and }\ DRx=D\pmatrix{15\\ 0}=\pmatrix{15\\ 0}, $$ so that $x^TPx=x^TD^2x=\|Dx\|^2=272>225=\|DRx\|^2=x^TR^TPRx=x^TRPR^Tx$. Therefore $P\not\preceq RPR^T$.