Acta mathematica scientia, Series B >
SOME PROPERTIES OF COMMUTING AND ANTI-COMMUTING m-INVOLUTIONS
Received date: 2011-11-01
Revised date: 2011-02-07
Online published: 2012-03-20
We define an m-involution to be a matrix K ∈ Cn×n for which Km = I. In this article, we investigate the class Sm (A) of m-involutions that commute with a diagonalizable matrix A ∈ Cn×n. A number of basic properties of Sm (A) and its related subclass Sm (A, X) are given, where X is an eigenvector matrix of A. Among them, Sm (A) is shown to have a torsion group structure under matrix multiplication if A has distinct eigenvalues and has non-denumerable cardinality otherwise. The constructive definition of Sm (A, X) allows one to generate all m-involutions commuting with a matrix with distinct eigenvalues. Some related results are also given for the class ˜ Sm (A) of m-involutions that anti-commute with a matrix A ∈ Cn×n.
Key words： Centrosymmetric; skew-centrosymmetric; bisymmetric; involution; eigenvalues
Mark Yasuda . SOME PROPERTIES OF COMMUTING AND ANTI-COMMUTING m-INVOLUTIONS[J]. Acta mathematica scientia, Series B, 2012 , 32(2) : 631 -644 . DOI: 10.1016/S0252-9602(12)60044-7
[1] Abu-Jeib I T. Involutions and Generalized Centrosymmetric and Skew-Centrosymmetric Matrices. Cana-dian Applied Mathematics Quarterly, 2007, 15
[2] Andrew A. Eigenvectors of certain matrices. Linear Algebra Appl, 1973, 7: 151–162
[3] Cantoni A, Butler P. Eigenvalues and eigenvectors of symmetric centrosymmetric matrices. Linear Algebra Appl, 1976, 13: 275–288
[4] Collar A. On centrosymmetric and centroskew matrices. Quart J Mech Appl Math, 1962, 15: 265–281
[5] Tao D, Yasuda M. A Spectral Characterization of Generalized Real Symmetric Centrosymmetric and Generalized Real Symmetric Skew-Centrosymmetric Matrices. SIAM J Matrix Anal Appl, 2002, 23: 885–895
[6] Trench W F. Characterization and properties of matrices with generalized symmetry or skew-symmetry. Linear Algebra Appl, 2004, 377: 207–218
[7] Trench W F. Characterization and properties of matrices with k-involutory symmetries. Linear Algebra Appl, 2008, 429: 2278–2290
[8] Trench W F. Characterization and properties of matrices with k-involutory symmetries II. Linear Algebra Appl, 2010, 432: 2782–2797
[9] Yasuda M. A spectral characterization of hermitian centrosymmetric and hermitian skew-centrosymmetric K-matrices. SIAM J Matrix Anal Appl, 2003, 25: 601–605
/
| 〈 |
|
〉 |