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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..2012-11-30
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    @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](http://books.google.com.hk/books?id=mlOa7wPX6OYC&pg=PA69&hl=zh-TW&source=gbs_toc_r&cad=4#v=onepage&q&f=false) or sec. 7.3 of Horn and Johnson's [Matrix Analysis](http://books.google.com.hk/books?id=PlYQN0ypTwEC&pg=PA411&hl=zh-TW&source=gbs_toc_r&cad=4#v=onepage&q&f=false). If the existence of SVD is not the question, why can't you just apply it?2012-11-30
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    user1551 thanks for the Help, I'll look it up!2012-12-01