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 positivedefinite. 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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decomposition, polar 
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