A Hilbert Space Problem Book pp 74-76 | Cite as

# 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.

## Keywords

Hilbert Space Extreme Point Positive Operator Correct Statement Polar Decomposition## Preview

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