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Math Help - commutative law of the product of two p vectors and pxp matrix

  1. #1
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    commutative law of the product of two p vectors and pxp matrix

    Hello,

    a and b are two column p-vectors
    A is a p \times p square matrix.

    Trying to proof something (see attachment) ... I had to force it a bit and even though it feels right, I do not know exactly how to apply commutative law on the following products so that they can cancel out with the division \frac{ab^TA^{-1}}{b^TA^{-1}a} I might be wrong but I think they both produce the same p \times p square matrix though looking at the original problem looks more like a scalar.

    Basically I need to arrive to \frac{ab^TA^{-1}}{ab^TA^{-1}} or \frac{b^TA^{-1}a}{b^TA^{-1}a}.

    I was trying something along the lines of (a)(b^TA^{-1})=(b^TA^{-1})^T(a)^T={A^{-1}}^Tba^T but I can't arrive to the desired result.

    Maybe I am applying the commutative rules wrongly ... btw I don't remember the rules name for this otherwise I would just look it up ..

    If they did cancel the output would be 1 or Identity_p?

    TIA,
    Best regards,
    bravegag

    This is the proof I am trying to make:

    commutative law of the product of two p vectors and pxp matrix-proofa.png
    Last edited by bravegag; April 24th 2010 at 06:42 AM.
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  2. #2
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    Already found the solution ... only needed to realize that e.g. b^TA^{-1}a is a scalar then I can move it around and solve the proof.
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