Question about Hermitian matrices

[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

 

[micromass]

What you're trying to prove is false. It won't be an inner product in general. You need $A$ to be positive-definite.

http://en.wikipedia.org/wiki/Positive-definite_matrix

 

[Bipolarity]

What you're trying to prove is false. It won't be an inner product in general. You need $A$ to be positive-definite.

http://en.wikipedia.org/wiki/Positive-definite_matrix

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

BiP

 

[micromass]

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

BiP

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

 

[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).