The Polar Decomposition of a Matrix

We prove that every square matrix (linear operator mapping the inner product space into itself) can be factored as the product of a unitary matrix and a positive semidefinite matrix. The factorization, A = S (A^T A)^{1/2} forces S to be a unitary mapping by the Singular Value Decomposition. The SVD plays an essential role in our proof. #mikethemathematician, #mikedabkowski, #profdabkowski