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Math Help - Inverse of this matrix

  1. #1
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    Inverse of this matrix

    Find the inverse of the matrix I_n+ab^T.

    I know that I'm really suppose to work a little bit on my own first before asking questions here... But I don't really know how to start this problem here other than just writing up the matrices in this manner - the (i,j) entry being 1+a_i^T*b_j where a_i^T is the ith row of a and  b_j is the jth column of b.

    Then the next question is finding the inverse of the matrix D+ab^T where D is a diagonal matrix.

    I suppose the first question will help me with this second?

    Thank you!!!
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  2. #2
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    Re: Inverse of this matrix

    Hey tttcomrader.

    Just to clarify, what are the dimensions of the matrices a and b (are they row/column or are they square)?
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  3. #3
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    Re: Inverse of this matrix

    The problem did not state, but I assume that a is n x m and b is m x n. Thanks!
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  4. #4
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    Re: Inverse of this matrix

    I would try and look at doing an eigen-decomposition and see if you can relate the eigen-decomposition of ab^t to that of ab^t + I.

    Remember that I = Q*Q^t = Q*Q^(-1) if Q is an orthogonal matrix.
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  5. #5
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    Re: Inverse of this matrix

    So I write ab^T=Q \Lambda Q^* then I have  (I-ab^T)^{-1}=(QQ^*-Q \Lambda Q^*)^{-1}=[Q(I- \Lambda )Q^*]^{-1} =Q^*(I- \Lambda)^{-1}Q

    So I know that the (i,j) entry of (I - \Lambda )^{-1} is  \frac {1}{1- \lambda _i}I \{ i=j \}

    Are there any ways I can simplify this? Thank you!
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  6. #6
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    Re: Inverse of this matrix

    I'm not sure you can: if you can diagonalize ab^t then you can get a specific solution.
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