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Math Help - Proof of trace of product of vector and matrices

  1. #1
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    Proof of trace of product of vector and matrices

    Hi, how can one prove the following identity?

    \mathrm{v^T} M^{-1} D M^{-1}\mathrm{v} =  \mathrm{trace }\left( (M^{-1}\mathrm{v}) (M^{-1}\mathrm{v}) ^T D\right),

    where \mathrm{v} is a vector, M is a positive definite and symmetric matrix, and D is a symmetric matrix, and dimension is n.

    Thank you for your help.
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  2. #2
    Super Member girdav's Avatar
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    Re: Proof of trace of product of vector and matrices

    Use the fact that v^TM^{-1}=(M^{-1}v)^T (because M^{-1} is symmetric) and that \operatorname{trace}(ABC)=\operatorname{trace}(CAB  ).
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  3. #3
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    Re: Proof of trace of product of vector and matrices

    Thank you, I got the identity now. We have to use also, in addition to the facts you mentioned, that the trace of a scalar equals the scalar, right? Thanks!
    (Now that the question is solved, how can I mark the post as such? Thanks again)
    Last edited by gilberto; June 14th 2012 at 04:09 PM. Reason: additional question about post
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  4. #4
    Super Member girdav's Avatar
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    Re: Proof of trace of product of vector and matrices

    That's right.

    (you can edit the title in the first post, and add "[Solved]" in)
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