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 $\displaystyle <cx,y> = c<x,y>$

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

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- Dec 2nd 2009, 12:11 AMwatersparkpositive 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 $\displaystyle <cx,y> = c<x,y>$

I'm having trouble with proving the rest. Any help will be greatly appreciated! - Dec 2nd 2009, 02:49 AMtonio

Well, the usual definition of positive definite Hermitian matrix is precisely that $\displaystyle <Ax,x>=x^{*}Ax>0\,\,\forall\,0\ne x\in V$ , with $\displaystyle x^{*}=$ the complex conjugate transpose of the vector $\displaystyle x$ , and from here you get positiviness.

$\displaystyle <x,ky>'=<Ax,ky>=(ky)^{*}Ax=\overline{k}y^{*}Ax=\ov erline{k}<Ax,y>$ , and you get skew-linearity...

Tonio