Companion matrices and their relations to Toeplitz and Hankel matrices

• Autorzy: Yousong Luo , Robin Hill
• Języki publikacji: EN

Abstrakty

EN

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.

Słowa kluczowe

EN

Companion matrix; Toeplitz matrix; Hankel matrix; Bezoutian

• Wydawca: De Gruyter Open
• Czasopismo: Special Matrices
• Rocznik: 2015
• Tom: 3
• Numer: 1

Daty

otrzymano

2015-04-22

zaakceptowano

2015-09-15

online

2015-10-07

Twórcy

autor

Yousong Luo

• Luo: School of Mathematical and Geospatial Sciences, RMIT University, GPO Box 2476V,
  Melbourne, Vic. 3001, AUSTRALIA

autor

Robin Hill

• Department of Electrical and Electronic Engineering, University of Melbourne, Vic. 3010, AUSTRALIA

Bibliografia

• [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).
• [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).
• [5] I. Gohberg and A. Semencul, On the inversion of finite Toeplitz matrices and their continuous analogs, Mat. Issled. 7 (2), 201-223 (1972).
• [6] Georg Heinig and Karla Rost, Introduction to Bezoutians, Operator Theory: Advances and Applications, Vol. 199, 25 - 118, (2010).
• [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).
• [11] Arthur Lim and Jialing Dai On product of companion matrices, Linear Algebra Appl., Vol. 435, Issue 11, 2921–2935, (2011). [WoS]
• [12] David G. Luenberger, Introduction to Dynamic Systems: Theory,Models, and Applications, JohnWiley&Sons, Inc., New York, (1979).

Identyfikatory

DOI

10.1515/spma-2015-0021