Polar Decomposition

Abstract

Every complex number is the product of a nonnegative number and a number of modulus 1; except for the number 0, this polar decomposition is unique. The generalization to finite matrices says that every complex matrix is the product of a positive matrix and a unitary one. If the given matrix is invertible, and if the order of the factors is specified (UP or PU), then, once again, this polar decomposition is unique. It is possible to get a satisfactory uniqueness theorem for every matrix, but only at the expense of changing the kind of factors admitted; this is a point at which partial isometries can profitably enter the study of finite-dimensional vector spaces. In the infinite-dimensional case, partial isometries are unavoidable. It is not true that every operator on a Hilbert space is equal to a product UP, with U unitary and P positive, and it does not become true even if U is required to be merely isometric. (The construction of concrete counterexamples may not be obvious now, but it will soon be an easy by-product of the general theory.) The correct statements are just as easy for transformations between different spaces as for operators on one space.