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October 31st, 2018, 07:54 PM   #1
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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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