KYUNGPOOK Math. J. 2019; 59(3): 505-513
Published online September 23, 2019
Copyright © Kyungpook Mathematical Journal.
A New Aspect of Comrade Matrices by Reachability Matrices
Maryam Shams Solary
Received: February 22, 2018; Revised: January 22, 2019; Accepted: January 26, 2019
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
Some recurrence relations from the general theory of orthogonal polynomials are shown in [1, 2, 4], these relations are given in terms of comrade matrices. Comrade matrices have regular form and may be used in finding the roots of a polynomial that an important problem in the numerical analysis.
As we know, Frobenius’s original idea used of companion matrices to find the zeros of a polynomial or a function. It can be expressed by some limitations and conditions for the condition number and floating point arithmetic [3, 4].
Specht, Boyd and Good et al. [5, 8] used this structure for finding the roots of a polynomial in Chebyshev form, for their method of rootfinding-by-proxy, and for Chebyshev interpolation. They derived these works using the Chebyshev-Frobenius matrix, which is also known as a colleague matrix.
The colleague matrix and companion matrix are known to be special cases of comrade matrices. In this paper, we try to use their algebraic structures to explain a connection between comrade matrices and reachability matrices. Using this connection, we then find the determinant, eigenvalues, and eigenvectors of their algebraic structures.
This paper is organized as follows: Some necessary details about comrade matrices and orthogonal polynomials are presented in Section 2. A connection between comrade matrices and reachability matrices is introduced in Section 3. Some results of this connection are given in Section 4. Finally, a summary is given in Section 5.
Assume, without loss of generality, that
The colleague matrix deduced from Chebyshev polynomials in
Comrade Matrices and Reachability Matrices
Let , . Then the matrix
The comrade matrix in
Now, we introduce an algebraic structure of the set
Let . We can easily see, is isomorphism with , then ideal ≺
For proof see Theorem 3.2. in this paper and Theorem 2.3 in .
Now we derive:
Now we derive the following theorem:
We try to show a connection between comrade matrices and reachability matrices. For this work, we introduce an algebraic structure and some properties of this structure. Then, using this structure, we find the determinant, eigenvalues, and eigenvectors of the reachability matrices. Also, we derive a factorization of the polynomial
- S. Barnett.
Congenial matrices. Linear Algebra Appl.., 41(1981), 277-298.
- DS. Bernstein. Matrix mathematics: theory, facts, and formulas,
, Princeton university press, Prenceton, 2009.
- JP. Boyd.
Computing real roots of a polynomial in Chebyshev series form through subdivision. Appl. Numer Math., 56(2006), 1077-1091.
- JP. Boyd.
Finding the zeros of a univariate equation: proxy rootfinders, Chebyshev interpolation, and the companion matrix. SIAM Rev.., 55(2)(2013), 375-396.
- JP. Boyd, and DH. Gally.
Numerical experiments on the accuracy of the ChebyshevFrobenius companion matrix method for finding the zeros of a truncated series of Chebyshev polynomials. J. Comput. Appl. Math., 205(2007), 281-295.
- G-S. Cheon, and H. Kim.
A new aspect of Hankel matrices via Krylov matrix. Linear Algebra Appl., 438(2013), 361-373.
- A. Ferrante, and HK. Wimmer.
Reachability matrices and cyclic matrices. Electron. J. Linear Algebra., 20(2010), 95-102.
- IJ. Good.
The colleague matrix, a Chebyshev analogue of the companion matrix. Quart J Math. Oxford Ser. (2), 12, 1961:61-68.
- Shams Solary. M. A connection between comrade matrices and reachability matrices. SIAM Conference on Applied Linear Algebra (LA15); 2015 p. 72-73.