The Singular Value Expansion for Arbitrary Bounded Linear Operators

Daniel K. Crane and Mark S. Gockenbach
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Daniel K. Crane: Department of Mathematical Sciences, Michigan Technological University, Houghton, MI 49931, USA
Mark S. Gockenbach: Department of Mathematical Sciences, Michigan Technological University, Houghton, MI 49931, USA

Mathematics, 2020, vol. 8, issue 8, 1-12

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.

Keywords: singular value expansion; inverse problems; regularization (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2020
References: View complete reference list from CitEc
Citations: View citations in EconPapers (1)

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