1. The problem statement, all variables and given/known data a) Explain how any square matrix A can be written as [itex] A = QS [/itex] where Q is orthogonal and S is symmetric positive semidefinite. b) Is it possible to write [itex] A = S_1 Q_1 [/itex] Where Q1 is orthogonal and S1 is symmetric positive definite? 2. Relevant equations [itex] A = U \Sigma V^T [/itex] 3. The attempt at a solution For a) I've gotten to the point where I've written: [itex] A = U V^T V \Sigma V^T [/itex] Which is just a rearrangement of the single value decomposition. From this I believe there is some logic as to why U V^T is orthogonal, and VΣV^T is symmetric positive definite, but I can't seem to figure out the reasoning. Any pointers? For b) I've surmised this is possible given that it is simply the "left polar decomposition" (http://en.wikipedia.org/wiki/Polar_decomposition) But again, I can't think about how to show this mathematically.