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 October 31st, 2018, 06:54 PM #1 Member   Joined: Feb 2018 From: Canada Posts: 46 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.
 November 30th, 2018, 05:29 PM #2 Member   Joined: Feb 2018 From: Canada Posts: 46 Thanks: 2 The (real) SVD of $A = U\Sigma V^{T}$ always exists. Inserting $I=V^{T}V$ in between $U$ and $\Sigma$, we find: $A = \underbrace{UV^{T}}_{=: Q}\underbrace{V\Sigma V^{T}}_{=:P},$ Since A is invertible, $A^{-1} = V\Sigma ^{-1}U^{T}$, which implies that $\Sigma$ is invertible as well. Since the singular values of A are positive, the matrix $V\Sigma V^{T}$ is symmetric positive-definite as it is a spectral decomposition with orthogonal eigenvectors and positive eigenvalues, and the product $UV^{T}$ is clearly orthogonal. We must find a symmetric $P$ and orthogonal $Q$. Thus: $A^{T}A = P^{T}Q^{T}QP = P^{T}P = P^2.$ Since A is invertible, $A^{T}A$ is symmetric positive-definite and it has a spectral decomposition $V\Lambda V^{T}$ with orthogonal eigenvectors. Thus $P = \sqrt{A^{T}A} = \sqrt{V\Lambda V^{T}} = V\Lambda^{\frac{1}{2}} V^{T},$ is also symmetric positive-definite. Finally, $Q = V\Lambda^{-\frac{1}{2}}V^{T}$ is orthogonal, since: \begin{aligned} Q^{T}Q &= V\Lambda^{-\frac{1}{2}}V^{T}A^{T}AV\Lambda^{-\frac{1}{2}}V^{T} = V\Lambda^{-\frac{1}{2}}V^{T}V\Lambda V^{T}V\Lambda^{-\frac{1}{2}}V^{T}, \\ &= V\Lambda^{-\frac{1}{2}}\Lambda \Lambda^{-\frac{1}{2}}V^{T} = VV^{T} = I. \end{aligned} Since Q is orthogonal: \begin{aligned} cond_2(A) &= \|A\|_2 \|A^{-1}\|_2 \\ &= \|QP\|_2 \|(QP)^{-1}\|_2 \\ &= \|QP\|_2 \|P^{-1}Q^{-1}\|_2 \\ &\leq\|Q\|_2\|P\|_2\|P^{-1}\|_2\|Q^{-1}\|_2 \\&= \|P\|_2\|Q\|_2\|Q^{-1}\|_2 |P^{-1}\|_2 \\ &= \|P\|_2\|P^{-1}\|_2 \\ &= cond_2(P) \end {aligned} On the other hand, $A = QP \rightarrow Q^{T}A = Q^{T}QP \rightarrow Q^{T}A = P$ \begin{aligned} \text{then} \quad cond_2(P) &= \|P\|_2 \|P^{-1}\|_2 \\ &= \|Q^{T}A\|_2 \|(Q^{T}A)^{-1}\|_2 \\ &= \|Q^{T}A\|_2 \|A^{-1}Q\|_2 \\ &\leq\|Q^{T}\|_2\|A\|_2\|A^{-1}\|_2\|Q\|_2 \\&= \|A\|_2\|Q\|_2\|Q^{-1}\|_2 |A^{-1}\|_2 \\ &= \|A\|_2\|A^{-1}\|_2 \\ & = cond_2(A) \end{aligned} Thus, $cond_2(A) = cond_2(P)$. Last edited by Shanonhaliwell; November 30th, 2018 at 05:31 PM.
 December 2nd, 2018, 07:36 PM #3 Senior Member   Joined: Sep 2016 From: USA Posts: 635 Thanks: 401 Math Focus: Dynamical systems, analytic function theory, numerics Your 1st post seems fine though its a bit of overkill. Note that if $A,B$ are linear operators, then $cond(AB) = cond(A)cond(B)$ which is trivial to prove. Then, since $cond(U) = 1$ for any unitary matrix, it follows that when $A = UP$, then $cond(A) = cond(P)$. In your 2nd post you seem to be done in the first expression so it isn't clear what you are trying to do. I am also mildly suspicious that your teacher has in mind you begin with an SVD since this is typically viewed as a generalized form of the polar decomposition. Thanks from Shanonhaliwell

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