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

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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

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