# Companion matrices and their relations to Toeplitz and Hankel matrices

Special Matrices (2015)

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

## Access Full Article

top## Abstract

top## How to cite

topYousong 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

top- [1] Yu. A. Al’pin and S. N. Il’in, Infinite extensions of Toeplitz matrices, J. Math. Sci. (N. Y.), Vol. 127, No. 3, 1957 - 1961, (2005).
- [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] Louis Brand, Companion matrix and its properties, Amer. Math. Monthly, Vol. 71, No. 6, 629 - 634, (1964).
- [4] Z. Cinkir, A fast elementary algorithm for computing the determinant of Toeplitz matrices, J. Comput. Appl. Math, Vol. 255, 353-361, (2014). Zbl1291.65142
- [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
- [6] Georg Heinig and Karla Rost, Introduction to Bezoutians, Operator Theory: Advances and Applications, Vol. 199, 25 - 118, (2010). Zbl1203.15020
- [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] 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] Thomas Kailath, Linear System, Prentice-Hall, Inc., (1980).
- [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] Arthur Lim and Jialing Dai On product of companion matrices, Linear Algebra Appl., Vol. 435, Issue 11, 2921–2935, (2011). [WoS] Zbl1222.39002
- [12] David G. Luenberger, Introduction to Dynamic Systems: Theory,Models, and Applications, JohnWiley&Sons, Inc., New York, (1979). Zbl0458.93001

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.