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Math Help - proving positive definiteness

  1. #1
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    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.
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  2. #2
    Junior Member
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    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, <br />
(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.
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