The Singular Value Expansion for Arbitrary Bounded Linear Operators

Daniel K Crane,Mark S Gockenbach

doi:10.3390/math8081346

Abstract

The singular value decomposition (SVD) is a basic tool for analyzing matrices. Regarding a general matrix as defining a linear operator and choosing appropriate orthonormal bases for the domain and co-domain allows the operator to be represented as multiplication by a diagonal matrix. It is well known that the SVD extends naturally to a compact linear operator mapping one Hilbert space to another; the resulting representation is known as the singular value expansion (SVE). It is less well known that a general bounded linear operator defined on Hilbert spaces also has a singular value expansion. This SVE allows a simple analysis of a variety of questions about the operator, such as whether it defines a well-posed linear operator equation and how to regularize the equation when it is not well posed.

Highlights

One of the most powerful ideas in linear algebra is diagonalization, which renders many problems completely transparent
The reader can compare these proofs to other treatments of the same results that can be found in the literature on inverse problems
In Groetsch’s monograph [14], the analysis is restricted to compact operators and Theorem 2.1.1, Corollary 3.1.2, and Theorem 3.2.2 correspond to our results; Groetsch’s proofs use the singular value expansion (2) for compact operators

Summary

Introduction

One of the most powerful ideas in linear algebra is diagonalization, which renders many problems completely transparent. If A ∈ Rm×n is a general matrix (not assumed to be symmetric or even square), the singular value decomposition (SVD) allows us to diagonalize the matrix, at the cost of using two different orthonormal bases. For a general (that is, not necessarily self-adjoint) compact operator T : X → Y, we can derive the singular value expansion (SVE) of T by applying the spectral theorem for self-adjoint compact operators to T ∗ T (see [3], Section 4.4). When T : X → Y is not necessarily compact, there still exists a singular value expansion in the following form.

The SVE of a Bounded Linear Operator

Relationship to the SVE of a Compact Operator

The SVE of Operators Related to T

Inverse Problems and the SVE

The SVE and Tikhonov Regularization

Discussion