The Canonical Forms of Permutation Matrices

被引：1

作者：

Li, Wen-Wei ^{[1
,2
]}

Hou, Xin ^{[3
]}

Wang, Qing-Wen ^{[4
]}

机构：

[1] Univ Sci & Technol China, Sch Math Sci, Hefei 230026, Peoples R China

[2] Anhui Int Studies Univ, Sch Informat & Math, Hefei 231201, Peoples R China

[3] Capital Normal Univ, Coll Elementary Educ, Beijing 100048, Peoples R China

[4] Shanghai Univ, Dept Math, Shanghai 200444, Peoples R China

来源：

SYMMETRY-BASEL | 2023年 / 15卷 / 02期

基金：

中国国家自然科学基金; 北京市自然科学基金;

关键词：

permutation matrix; monomial matrix; permutation similarity; canonical form; cycle matrix decomposition; cycle factorization;

D O I：

10.3390/sym15020332

中图分类号：

O [数理科学和化学]; P [天文学、地球科学]; Q [生物科学]; N [自然科学总论];

学科分类号：

07 ; 0710 ; 09 ;

摘要：

We address classification of permutation matrices, in terms of permutation similarity relations, which play an important role in investigating the reducible solutions of some symmetric matrix equations. We solve the three problems. First, what is the canonical form of a permutation similarity class? Second, how to obtain the standard form of arbitrary permutation matrix? Third, for any permutation matrix A, how to find the permutation matrix T, such that T(-1 )AT is in canonical form? Besides, the decomposition theorem of permutation matrices and the factorization theorem of both permutation matrices and monomial matrices are demonstrated.

引用

页数：14

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