1. Aug 16, 2013

Bipolarity

I am trying to prove that for any two vectors x,y in $ℂ^{n}$ the product $\langle x,y \rangle = xAy^{*}$ is an inner product where $A$ is an $n \times n$ Hermitian matrix.

This is actually a generalized problem I created out of a simpler textbook problem so I am not even sure if this is true although I believe it is true.

I proved most of the axioms for an inner product space, except the axiom that $\langle x,x \rangle > 0$ if $x ≠ 0$. This is giving me trouble, since I first had to prove that $<x,x>$ is real (which I have done) but am still having trouble to actually prove that it is positive.

Any tips? Thanks!

BiP

Last edited: Aug 16, 2013
2. Aug 16, 2013

micromass

Staff Emeritus
3. Aug 16, 2013

Bipolarity

Ah I see micro! I hadn't gotten there yet in my coursework. Could you give me a counterexample please? thanks.

BiP

4. Aug 16, 2013

micromass

Staff Emeritus
It should be very easy for you to find a counterexample. You can even look for a $1\times 1$-matrix as a counterexample.

5. Aug 16, 2013

chiro

Positive definite matrices are defined to give the property you are seeking.

By their definition, xAx' > 0 for all vectors x if A is positive definite and these matrices act like norms and metrics in the way they transform vectors (and are used in situations that have this property like inner products in geometry and variance/co-variance in probability/statistics).