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Linear AlgebraHardSingular Values & the SVD

Singular Values & the SVD · Hard problem

Every square matrix has a polar decomposition A=QSA = QS with QQ orthogonal and S=ATAS = \sqrt{A^{T}A} symmetric positive semidefinite; from the SVD, S=VΣVTS = V\Sigma V^{T}. For A=(3045)A = \begin{pmatrix}3 & 0\\4 & 5\end{pmatrix}, find the entry of SS in row 1, column 2. Give your answer in simplest radical form.