# A New Aspect of Comrade Matrices by Reachability Matrices

• Solary, Maryam Shams (Department of Mathematics, Payame Noor University)
• Accepted : 2019.01.26
• Published : 2019.09.23
• 13 2

#### Abstract

In this paper, we study orthanogonal polynomials by looking at their comrade matrices and reachability matrices. First, we focus on the algebraic structure that is exhibited by comrade matrices. Then, we explain some properties of this algebraic structure which helps us to find a connection between comrade matrices and reachability matrices. In the last section, we use this connection to determine the determinant, eigenvalues, and eigenvectors of these matrices. Finally, we derive a factorization for det R(A, x), where R(A, x) is the reachability matrix for a comrade matrix A and x is a vector of indeterminates.

#### Keywords

Orthogonal polynomials;companion matrix, comrade matrix, reachability matrix

#### References

1. S. Barnett, Congenial matrices, Linear Algebra Appl., 41(1981), 277-298. https://doi.org/10.1016/0024-3795(81)90104-X
2. D. S. Bernstein, Matrix mathematics: theory, facts, and formulas, Princeton university press, Prenceton, 2009.
3. J. P. Boyd, Computing real roots of a polynomial in Chebyshev series form through subdivision, Appl. Numer. Math., 56(2006), 1077-1091. https://doi.org/10.1016/j.apnum.2005.09.007
4. J. P. Boyd, Finding the zeros of a univariate equation: proxy rootfinders, Chebyshev interpolation, and the companion matrix, SIAM Rev., 55(2)(2013), 375-396. https://doi.org/10.1137/110838297
5. J. P. Boyd and D. H. Gally, Numerical experiments on the accuracy of the Chebyshev-Frobenius companion matrix method for finding the zeros of a truncated series of Chebyshev polynomials, J. Comput. Appl. Math., 205(2007), 281-295. https://doi.org/10.1016/j.cam.2006.05.006
6. G.-S. Cheon and H. Kim, A new aspect of Hankel matrices via Krylov matrix, Linear Algebra Appl., 438(2013) 361-373. https://doi.org/10.1016/j.laa.2012.07.040
7. A. Ferrante and H. K. Wimmer, Reachability matrices and cyclic matrices, Electron. J. Linear Algebra, 20(2010), 95-102.
8. I. J. Good, The colleague matrix, a Chebyshev analogue of the companion matrix, Quart. J. Math. Oxford Ser. (2), 12(1961), 61-68. https://doi.org/10.1093/qmath/12.1.61
9. M. Shams Solary, A connection between comrade matrices and reachability matrices, SIAM Conference on Applied Linear Algebra (LA15), (2015), 72-73.