# Matrix existence..

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?