SOME PROPERTIES OF COMMUTING AND ANTI-COMMUTING m-INVOLUTIONS

doi:10.1016/S0252-9602(12)60044-7

数学物理学报(英文版) ›› 2012, Vol. 32 ›› Issue (2): 631-644.doi: 10.1016/S0252-9602(12)60044-7

SOME PROPERTIES OF COMMUTING AND ANTI-COMMUTING m-INVOLUTIONS

Mark Yasuda

9525 Compass Point Drive South, San Diego, CA 92126, U.S.A.

收稿日期:2011-11-01 修回日期:2011-02-07 出版日期:2012-03-20 发布日期:2012-03-20

SOME PROPERTIES OF COMMUTING AND ANTI-COMMUTING m-INVOLUTIONS

Mark Yasuda

9525 Compass Point Drive South, San Diego, CA 92126, U.S.A.

Received:2011-11-01 Revised:2011-02-07 Online:2012-03-20 Published:2012-03-20

摘要/Abstract

摘要：

We define an m-involution to be a matrix K ∈ C^n×nfor which K^m= I. In this article, we investigate the class Sm (A) of m-involutions that commute with a diagonalizable matrix A ∈ C^n×n. A number of basic properties of S_m (A) and its related subclass S_m (A, X) are given, where X is an eigenvector matrix of A. Among them, S_m (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 S_m (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 ˜ S_m (A) of m-involutions that anti-commute with a matrix A ∈ C^n×n.

关键词: Centrosymmetric, skew-centrosymmetric, bisymmetric, involution, eigenvalues

Abstract:

We define an m-involution to be a matrix K ∈ C^n×nfor which K^m= I. In this article, we investigate the class S_m (A) of m-involutions that commute with a diagonalizable matrix A ∈ C^n×n. A number of basic properties of S_m (A) and its related subclass S_m (A, X) are given, where X is an eigenvector matrix of A. Among them, S_m (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 S_m (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 ˜ S_m (A) of m-involutions that anti-commute with a matrix A ∈ C^n×n.

Key words: Centrosymmetric, skew-centrosymmetric, bisymmetric, involution, eigenvalues

中图分类号:

15A18

Mark Yasuda. SOME PROPERTIES OF COMMUTING AND ANTI-COMMUTING m-INVOLUTIONS[J]. 数学物理学报(英文版), 2012, 32(2): 631-644.

Mark Yasuda. SOME PROPERTIES OF COMMUTING AND ANTI-COMMUTING m-INVOLUTIONS[J]. Acta mathematica scientia,Series B, 2012, 32(2): 631-644.

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SOME PROPERTIES OF COMMUTING AND ANTI-COMMUTING m-INVOLUTIONS

SOME PROPERTIES OF COMMUTING AND ANTI-COMMUTING m-INVOLUTIONS

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