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Math Help - Matrix Analysis - Horn Question

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    Matrix Analysis - Horn Question



    Question is in regards to 6 (with reference to 4).

    I get the hint, but am not seeing how some principal matrix being non-singular helps to prove rank of A-rI is n-1.
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    Re: Matrix Analysis - Horn Question

    What you have to show is that if B is a n\times n matrix which is not invertible and such that a principal submatrix of size (n-1)\times (n-1) is invertible then the rank of B is n-1. After permuting if necessary the rows and the columns, you can assume that it's the submatrix after deleting the n-th row and n-th column. Let B' the submatrix, and consider the blockwise matrix C=\begin{pmatrix}B'&0\\ 0&1\end{pmatrix}. Multipliying by C, we can see that the rank of B is at least n-1. It can't be more since B is not invertible.
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    Re: Matrix Analysis - Horn Question

    I understand what I'm meant to show; not sure I follow your proof to be honest (syntax is a bit awkward).

    Jumped on Wiki and "determinantal rank" jumped out; that seems like the easiest route with n-1 being largest possible rank of principle submatrix.
    Last edited by ANDS!; June 15th 2012 at 03:19 AM.
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