A matrix inequality between two positive definite matrices and a positive scalar.

Question

I am trying to prove the following inequality:

$$ P R^{-1} P - 2 \epsilon P + \epsilon^2 R \geq 0, $$

where matrices $P$ and $R$ are definite positive, and $\epsilon$ is a real-valued positive scalar. It seems like the inequality always holds. I`m just wondering why.

Answer 1

If you let $S= (\sqrt{R})^{-1}$ and pre & post multiply by $S$ we get $(SPS)^2 - 2 \epsilon SPS + \epsilon^2 I = (SPS - \epsilon I)^2$ which is positive semi definite.