# positive definite matrix and inner products

• Dec 2nd 2009, 12:11 AM
waterspark
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 $\displaystyle <cx,y> = c<x,y>$

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

Originally Posted by waterspark
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!

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