# Homework Help: Second Hermitian scalar product

1. Dec 7, 2008

### Ioiô

1. The problem statement, all variables and given/known data

Let (u,v)1 be a second Hermitian scalar product on a vector space V.

Claim: There exists a positive transformation T with respect to the given scalar product (u,v) such that (u,v)1 = (Tu,v) for all u,v in V.

2. Relevant equations

A transformation T is positive if

2) (Tv, v) greater than or equal to zero (and real, of course)

3. The attempt at a solution

I am confused at what we are trying to prove. Is T positive with respect to (u,v) or are we using the definition of (u,v)1 and showing it is positive?

Initially, my plan to prove the existence of T was to first choose a matrix with respect to some orthonormal basis and then show that T is positive with respect to the inner product (u,v)1. However, as mentioned, the claim is that T is positive with respect to (u,v). How does (u,v)1 fit into the problem? (What is meant by second "Hermitian product?")

Thanks