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Math Help - would like help on formulating proof

  1. #1
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    would like help on formulating proof

    Dear all,

    I have an exercise which asks me to prove the following;

    Let A be a non-singular matrix with transpose  A^T and inverse  A^{-1}
    Let b be a vector with transpose  b^T .

    Prove that for b^T A^{-1} b =//not equal to//=  -1

     (A + b b^T)^{-1} = A^{-1} - (A^{-1} b b^T A^{-1})/(1+b^T A^{-1} b)


    Regards,
    Reinout

    Last edited by reinoutg; January 27th 2014 at 05:34 AM.
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  2. #2
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    Re: would like help on formulating proof

    Quote Originally Posted by reinoutg View Post
    Dear all,

    I have an exercise which asks me to prove the following;

    Let A be a non-singular matrix with transpose A' and inverse inv(A)
    Let b be a vector with transpose b'.

    Prove that for b' A b =//not equal to//= -1

    inv( A + b b' ) = inv(A) - inv(A) b b' inv(A) / ( 1 + b inv(A) b)

    I tried to give the easiest notation possible and hope you can help me.

    Regards,
    Reinout

    p.s. I attached an image to clarify the equation.
    there's a problem in your expression.

    you have 1+b A^{-1}b in the denominator.

    b$ is $n\times 1

    A^{-1}$ is $n\times n

    $ so $A^{-1}b$ is $n\times 1$ and cannot be multiplied on the left by a $n \times 1$ vector.$

    do you maybe mean 1+b^T A^{-1} b ?
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  3. #3
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    Re: would like help on formulating proof

    Thank you for replying romsek,

    I had not noticed this yet, but it must be a typo in the exercise. This indeed makes it either 1+b^T A^{-1} b or 1+b A^{-1} b^T.
    I am also dubious about the  b^TAb is not equal to -1 since obviously it is there to make the denominator be unequal to 0.
    Therefore I suppose this is another typo which is supposed to be b^TA^{-1}b.
    I will makes these two alterations in the original post.
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