Singular Value Decomposition (SVD) and Polar Form

Abstract

In this section we assume that we are dealing with a real Euclidean space E. Let $f : E \rightarrow E$ be any linear map. In general, it may not be possible to diagonalize f. We show that every linear map can be diagonalized if we are willing to use two orthonormal bases. This is the celebrated singular value decomposition (SVD). A close cousin of the SVD is the polar form of a linear map, which shows how a linear map can be decomposed into its purely rotational component (perhaps with a flip) and its purely stretching part.