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Math Help - positive definite matrix and inner products

  1. #1
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    positive definite matrix and inner products

    I'm trying to prove that if A is a positive definite Hermitian matrix, <x,y>' = <Ax,y> defines an inner product.

    So far I've only got <cx,y> = c<x,y>

    I'm having trouble with proving the rest. Any help will be greatly appreciated!
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  2. #2
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    Quote Originally Posted by waterspark View Post
    I'm trying to prove that if A is a positive definite Hermitian matrix, <x,y>' = <Ax,y> defines an inner product.

    So far I've only got <cx,y> = c<x,y>

    I'm having trouble with proving the rest. Any help will be greatly appreciated!

    Well, the usual definition of positive definite Hermitian matrix is precisely that <Ax,x>=x^{*}Ax>0\,\,\forall\,0\ne x\in V , with x^{*}= the complex conjugate transpose of the vector x , and from here you get positiviness.
    <x,ky>'=<Ax,ky>=(ky)^{*}Ax=\overline{k}y^{*}Ax=\ov  erline{k}<Ax,y> , and you get skew-linearity...

    Tonio
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