# Proof that these Hessian matrix identities are similar matrices

I am wondering if $Q, P$ are similar matrices where for a function

$f:\mathbb{R^n}\to\mathbb{R}$ and for a diagonal matrix $D$

$Q=I-D^{-1}\nabla^2f(x)$ and $P=I-D^{-1/2}\nabla^2f(x)D^{-1/2}$.

Similar matrix definition: $A,B$ are similar if $A=P^{-1}BP$ for some $P$.

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Let's assume $$P=X^{-1}QX,$$ so that $P$ and $Q$ are similar.
What do you then get if you set $X=D^{-1/2}$?