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Displacements of matrix products

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Book cover Applied Algebra, Algebraic Algorithms and Error-Correcting Codes (AAECC 1995)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 948))

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Abstract

For fixed matrices M and N, either of the linear transformations A ↦ A-MAN or or A ↦ MA-AN is called a displacement of the matrix A. Displacement can greatly reduce the rank of structured matrices, such as circulant, Vandermonde, Toeplitz and Hankel matrices. This rank reduction has been widely used for inverting structured matrices. In this paper, several formulas are given for both types of displacements applied to matrix products. Very few results for matrix products are known, yet they are desirable for dealing with matrix equations such as P 2=P, AA *=I, and A=U∑V *.

This research was partially supported by a grant from a Science & Engineering Fellowship of the University of Delaware Honors Program.

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References

  1. J. Chun and T. Kailath. Displacement structure for Hankel, Vandermonde, and related (derived) matrices. Linear Algebra and Its Applications, 151:199–227, 1991.

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  2. Georg Heinig and Karla Rost. Algebraic Methods for Toeplitz-like Matrices and Operators. Birkhäuser Verlag, Boston, 1984.

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  3. Victor Pan. On computations with dense structured matrices. Mathematics of Computation, 55:179–190, 1990.

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  4. David H. Wood. Product rules for the displacement of near-Toeplitz matrices. Linear Algebra and Its Applications, 188:641–663, 1993.

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  5. S. Barnett and M. J. C. Gover. Some extensions of Hankel and Toeplitz matrices. Linear and Multilinear Algebra, 14:45–65, 1983.

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Author information

Author notes

  1. Quyen L. Nguyen

    Present address: Computer Science Division, University of California at Berkeley, 94720-0001, Berkeley, CA

Authors and Affiliations

  1. Department of Computer and Information Sciences, University of Delaware, 19716, Newark, DE, USA

    Quyen L. Nguyen & David H. Wood

Authors
  1. Quyen L. Nguyen
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  2. David H. Wood
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Editor information

Gérard Cohen Marc Giusti Teo Mora

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© 1995 Springer-Verlag Berlin Heidelberg

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Nguyen, Q.L., Wood, D.H. (1995). Displacements of matrix products. In: Cohen, G., Giusti, M., Mora, T. (eds) Applied Algebra, Algebraic Algorithms and Error-Correcting Codes. AAECC 1995. Lecture Notes in Computer Science, vol 948. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-60114-7_29

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  • DOI: https://doi.org/10.1007/3-540-60114-7_29

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-60114-2

  • Online ISBN: 978-3-540-49440-9

  • eBook Packages: Springer Book Archive

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