My Math Forum Polar decomposition

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 October 31st, 2018, 07:54 PM #1 Member   Joined: Feb 2018 From: Canada Posts: 37 Thanks: 2 Polar decomposition I am having trouble with the fist part of this question. The polar decomposition of a real invertible matrix $A\in R^{n*n}$ is: $A=QP$ where $Q \in R^{n*n}$ is orthogonal and $P\in R^{n*n}$ symmetric positive-definite. Prove that it always exist. What is $cond_2(A)$ in terms of its polar factors $(Q \text{ and } P)$? For the second part, $cond_2(A) = \|A\|_2\|A^{-1}\|_2 = \|QP\|_2\|P^{-1}Q^{-1}\|_2 \leq \|Q\|_2\|P\|_2\|P^{-1}\|_2\|Q^{-1}\|_2 = \|P\|_2\|P^{-1}\|_2 \text{ since Q is orthogonal}$ Can someone hep me with the first part. I think we can start with singular value decomposition as $A = U\Sigma V^{T}$ then transform this into the polar decomposition.

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