# Matrix inequality: conjugating positive matrix by $R<-I$

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

• by $R\lt I$ do you mean $R-I$ is negative definite? – Ellya May 4 '14 at 20:31

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 $P\not\preceq RPR^T$.