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The Singular Value Expansion for Arbitrary Bounded Linear Operators

Author

Listed:
  • 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
    Current address: Department of Mathematical Sciences, University of Delaware, Newark, DE 19716, USA.)

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.

Suggested Citation

  • Daniel K. Crane & Mark S. Gockenbach, 2020. "The Singular Value Expansion for Arbitrary Bounded Linear Operators," Mathematics, MDPI, vol. 8(8), pages 1-12, August.
    • Handle: RePEc:gam:jmathe:v:8:y:2020:i:8:p:1346-:d:397952
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    Cited by:

    1. Mariya Kornilova & Vladislav Kovalnogov & Ruslan Fedorov & Mansur Zamaleev & Vasilios N. Katsikis & Spyridon D. Mourtas & Theodore E. Simos, 2022. "Zeroing Neural Network for Pseudoinversion of an Arbitrary Time-Varying Matrix Based on Singular Value Decomposition," Mathematics, MDPI, vol. 10(8), pages 1-12, April.

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