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Second Hermitian scalar product

  1. Dec 7, 2008 #1
    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

    1) T is self-adjoint
    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
     
  2. jcsd
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