Chapter 7, Problem 27SE

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 = UΣV^T, and observe that A = (UΣU^T)(UV^T).] 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.