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Math Help - Inner Product Properties

  1. #1
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    Inner Product Properties

    Let V = Mnn be the real vector space of all n x n matrices. If A and B
    are in V, we define (A,B) = Tr(B^TA) where Tr stands for trace and T
    stands for transpose. Prove that this function is an inner product on V.

    I don't understand how I would prove that the function is an inner
    product.

    According to my book, an inner product must satisfy the following
    properties:

    a) (u,u) > 0; (u,u) = 0 iff u = 0v.
    b) (v,u) = (u,v) for any u,v in V.
    c) (u+v,w) = (u,w) + (v,w) for any u,v,w in V.
    d) (cu,v) = c(u,v) for any u,v in V and c is a real scalar.

    I plugged in u = Tr(BTA).
    I get how I can prove the first property.
    (u,u) would be > 0 because it is a square. And if u = 0, then (u,u) = 0.
    I don't get how I would prove the other properties. What would I plug
    in v as? Should A and B be u and v, or is there another way? Thank you
    in advance!
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  2. #2
    Moo
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    Hello,

    Let's rephrase the properties with the notations your text uses =)

    a) (A,B) > 0; (A,A) = 0 iff A = 0B. << be careful, it's iff, not if
    b) (B,A) = (A,B) for any A,B in V.
    c) (A+B,C) = (A,C) + (B,C) for any A,B,C in V.
    d) (mA,B) = m(A,B) for any A,B in V and m is a real scalar.

    A, B and C are matrices.

    For example ({\color{blue}A+B},{\color{red}C})=\text{Tr}({\col  or{red}C}^T({\color{blue}A+B}))

    Use basic properties such as linearity of the trace to prove them
    Try to do them, and even if you have a slight doubt, don't hesitate to post it here.
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  3. #3
    MHF Contributor arbolis's Avatar
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    b) (v,u) = (u,v) for any u,v in V.
    Strange... I've learned that (v,u) must equals (\overline{u,v}), but I agree that when v and u takes their values in \mathbb{R} it makes no difference.
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  4. #4
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    Quote Originally Posted by arbolis View Post
    Strange... I've learned that (v,u) must equals (\overline{u,v}), but I agree that when v and u takes their values in \mathbb{R} it makes no difference.
    That is for a complex inner product space.
    In a real inner product space that is okay.
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  5. #5
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    That explained everything! Thanks so much!
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