Companion matrices and their relations to Toeplitz and Hankel matrices

Yousong Luo; Robin Hill

Special Matrices (2015)

  • Volume: 3, Issue: 1, page 214-226, electronic only
  • ISSN: 2300-7451

Abstract

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In this paper we describe some properties of companion matrices and demonstrate some special patterns that arisewhen a Toeplitz or a Hankel matrix is multiplied by a related companion matrix.We present a necessary and sufficient condition, generalizing known results, for a matrix to be the transforming matrix for a similarity between a pair of companion matrices. A special case of our main result shows that a Toeplitz or a Hankel matrix can be extended using associated companion matrices, preserving the Toeplitz or Hankel structure respectively.

How to cite

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Yousong Luo, and Robin Hill. "Companion matrices and their relations to Toeplitz and Hankel matrices." Special Matrices 3.1 (2015): 214-226, electronic only. <http://eudml.org/doc/275832>.

@article{YousongLuo2015,
abstract = {In this paper we describe some properties of companion matrices and demonstrate some special patterns that arisewhen a Toeplitz or a Hankel matrix is multiplied by a related companion matrix.We present a necessary and sufficient condition, generalizing known results, for a matrix to be the transforming matrix for a similarity between a pair of companion matrices. A special case of our main result shows that a Toeplitz or a Hankel matrix can be extended using associated companion matrices, preserving the Toeplitz or Hankel structure respectively.},
author = {Yousong Luo, Robin Hill},
journal = {Special Matrices},
keywords = {Companion matrix; Toeplitz matrix; Hankel matrix; Bezoutian; companion matrix},
language = {eng},
number = {1},
pages = {214-226, electronic only},
title = {Companion matrices and their relations to Toeplitz and Hankel matrices},
url = {http://eudml.org/doc/275832},
volume = {3},
year = {2015},
}

TY - JOUR
AU - Yousong Luo
AU - Robin Hill
TI - Companion matrices and their relations to Toeplitz and Hankel matrices
JO - Special Matrices
PY - 2015
VL - 3
IS - 1
SP - 214
EP - 226, electronic only
AB - In this paper we describe some properties of companion matrices and demonstrate some special patterns that arisewhen a Toeplitz or a Hankel matrix is multiplied by a related companion matrix.We present a necessary and sufficient condition, generalizing known results, for a matrix to be the transforming matrix for a similarity between a pair of companion matrices. A special case of our main result shows that a Toeplitz or a Hankel matrix can be extended using associated companion matrices, preserving the Toeplitz or Hankel structure respectively.
LA - eng
KW - Companion matrix; Toeplitz matrix; Hankel matrix; Bezoutian; companion matrix
UR - http://eudml.org/doc/275832
ER -

References

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  2. [2] D. Bini, V. Pan, Eflcient algorithms for the evaluation of the eigenvalues of (block) banded Toeplitz matrices, Math. Comp. 50, 431-448, (1988). Zbl0646.65035
  3. [3] Louis Brand, Companion matrix and its properties, Amer. Math. Monthly, Vol. 71, No. 6, 629 - 634, (1964). 
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  5. [5] I. Gohberg and A. Semencul, On the inversion of finite Toeplitz matrices and their continuous analogs, Mat. Issled. 7 (2), 201-223 (1972). Zbl0288.15004
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  7. [7] Georg Heinig and Karla Rost, Algebraic methods for Toeplitz-like matrices and operators, Operator Theory: Advances and Applications, Vol. 13, Birkhäuser Verlag, Basel, (1984). 
  8. [8] R. Hill, Y. Luo and U. Schwerdtfeger, Exact solutions to a two-block l1 optimal control problem, Porceedings of the 7th IFAC Symposium on Robust Control Design, ROCOND’12, Jakob Stoustrup, Elsevier, United Kingdom, pp. 461-466 ( 2012) 
  9. [9] Thomas Kailath, Linear System, Prentice-Hall, Inc., (1980). 
  10. [10] F. I. Lander, The Bezoutian and the inversion of Hankel and Toeplitz matrices (in Russian),Mat. Issled., Vol. 9, 69–87, (1974). Zbl0331.15017
  11. [11] Arthur Lim and Jialing Dai On product of companion matrices, Linear Algebra Appl., Vol. 435, Issue 11, 2921–2935, (2011). [WoS] Zbl1222.39002
  12. [12] David G. Luenberger, Introduction to Dynamic Systems: Theory,Models, and Applications, JohnWiley&Sons, Inc., New York, (1979). Zbl0458.93001

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