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How to prove that for any matrix $A\in \mathbb R^{m\times n}$ ($m\geq n$) such that $rank(A)=r$ there exists a nonsingular matrix $P$ and an orthogonal matrix $U$ such that, \begin{align*} A=U\Gamma P^{-1}, \end{align*} where, \begin{align*} \displaystyle\Gamma=\left(\begin{array}{cc} \textrm{diag}(\gamma_1, \ldots, \gamma_r)&0\\ 0&0 \end{array}\right), \end{align*} and, \begin{align*} \gamma_i=\sqrt{p_i^TA^TAp_i},\ i=1, \ldots, r. \end{align*} If those matrices indeed exist I can prove the equality for $\gamma_i$ using SVD, but I wasn't manage to show they really exist..

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I am not sure if I understand your question correctly. However, if you are not questioning the existence of SVD, then the SVD of $A$ already gives you the required decomposition, doesn't it?

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I thought that too @user1551 but it would be strange not to ask for proving the SVD decomposition directly, this question is a bit strange.. –  PtF Nov 30 '12 at 15:12
    
@Dion: I don't understand. If you are looking for a proof of the existence of SVD, you may look up any popular reference book. See, e.g. sec. 2.5 of Golub and van Loan's Matrix Computations or sec. 7.3 of Horn and Johnson's Matrix Analysis. If the existence of SVD is not the question, why can't you just apply it? –  user1551 Nov 30 '12 at 15:46
    
user1551 thanks for the Help, I'll look it up! –  PtF Dec 1 '12 at 15:23
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