Document Type

Article

Publication Date

8-1-2020

Department

Department of Mathematical Sciences

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.

Publisher's Statement

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/). Publisher’s version of record: https://doi.org/10.3390/MATH8081346

Publication Title

Mathematics

Creative Commons License

Creative Commons Attribution 4.0 License
This work is licensed under a Creative Commons Attribution 4.0 License.

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Publisher's PDF

Included in

Mathematics Commons

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