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Prove that any n × n matrix A admits a polar decomposition of the form A = PQ, where P is an n × n positive semidefinite matrix with the same rank as A and where Q is an n × n orthogonal matrix. [Hint: Use a singular value decomposition, A = UΣVT, and observe that A = (UΣUT)(UVT).] This decomposition is used, for instance, in mechanical engineering to model the deformation of a material. The matrix P describes the stretching or compression of the material (in the directions of the eigenvectors of P), and Q describes the rotation of the material in space.
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