1. ## proving positive definiteness

Can someone help with this exercise?
Given a symmetric n x n positive definite matrix A, and an arbitrary non-
singular n x n matrix P, show that P'AP is positive definite.

2. We know that $(Ax,x)\geq0, \forall x\in V$, $A'=A$, and $(Ax,x)=0 \implies x=0$.

So we want to show P'AP>0.

Let $x\in V,
(P'APx,x)=(A(Px),(Px))\geq0$
, since $Px \in V$
Similarly, $(A(Px),(Px))=0 \implies Px=0$ and since P is invertible this implies $x=0$.

Now self-adjoint should be given probably from a theorem, but here's a proof.
If A=A', then (P'AP)'=P'A'P''=P'AP.