四元数矩阵方程AXA~H+B~HYB=C的埃米特解分量极秩

英文篇名：Extremal Ranks of Complex Components in Hermite Solutions of the Matric Equation AXA~H+B~HYB=C over Quaternion Field
作者：连德忠 ; 谢锦山 ; 李美莲 ; 游德有 ; 吴敏丽
英文作者：LIAN Dezhong;XIE Jinshan;LI Meilian;YOU Deyou;WU Minli;School of Mathematics and Information Engineering,Longyan University;
关键词：四元数 ; 矩阵方程 ; 复表示 ; 埃米特解 ; 分块矩阵 ; 极秩
英文关键词：quaternion;;matric equations;;complex representation;;Hermite solution;;block matrix;;extremal ranks
中文刊名：FDXB
英文刊名：Journal of Fudan University(Natural Science)
机构：龙岩学院数学与信息工程学院;
出版日期：2019-02-15
出版单位：复旦学报(自然科学版)
年：2019
期：v.58
基金：国家自然科学基金(11601214,11526107);; 福建省教育厅课程思政项目(KC18084);福建省教育厅重点项目(JA14299);; 福建省自然科学基金(2015J05010);; 福建省高校杰出青年科研人才支持项目;; 龙岩学院科研项目(LG2014001,LB2014018)
语种：中文;
页：FDXB201901004
页数：9
CN：01
ISSN：31-1330/N
分类号：29-37

摘要

借助四元数矩阵的复表示方式Φ(·),将四元数体上的线性矩阵方程AXAH+BHYB=C转换为复数域上的等价复矩阵方程Φ(A)X～(Φ(A))H+(Φ(B))HY～Φ(B)=Φ(C).同时,利用复矩阵方程的埃米特解和分块矩阵的极秩性质,求出原方程埃米特通解中复矩阵分量集{X0},{X1},{Y0}和{Y1}的最大秩、最小秩公式.作为这些极秩公式的应用,最后推导出原方程埃米特通解中包含复矩阵解或全为复矩阵解的充要条件.
By using a complex representation of quaternion matrixΦ(·),the linear matrix equation AXAH+BHYB=C～over quaternion field is changed into the matrix equationΦ(A)X[Φ(A)]H+[Φ(B)]H～YΦ(B)=Φ(C)over complex field.Then,by using the Hermite solutions of this complex matrix equation and many properties about extreme ranks of block matrix,the formulas of the extreme ranks of complex matrices{X0},{X1},{Y0},{Y1}are obtained.These complex matrices are the complex components of the Hermite solutions X =X0 +X1 j,Y=Y0+Y1 j of the quaternion matrix equation.As an application,we give the necessary and sufficient conditions for the following～special cases:(a)There is at least a pair of complex matrices{X0,～Y0}is the Hermite solution of the quaternion matrix equation;(b)All matrices in the Hermite solutions of the quaternion matrix equation are complex.

引文

[1]HUNGERFORD T W.Algebra[M].New York,USA:Spring-Verlag,1980.
    [2]ZHANG F.Quaternions and matrices of quaternions[J].Linear Algebra and Its Applications,1997,251:21-57.
    [3]王卿文,薛有才.体和环上的矩阵方程[M].北京:知识出版社,1996:167-183.
    [4]WANG Q W,ZHANG H S,YU S W.On solutions to the quaternion matrix equation AXB+CYD=E[J].Electronic Journal of Linear Algebra,2008,17:343-358.
    [5]JIANG T S,CHEN L.Algebraic algorithms for least squares problem in quaternionic quantum theory[J].Computer Physics Communications,2007,176(7):481-485.
    [6]JIANG T S,ZHAO J L,WEI M S.A new technique of quaternion equality constrained least squares problem[J].Journal of Computational and Applied Mathematics,2008,216(2):509-513.
    [7]TIAN Y G.Completing triangular block matrices with maximal and minimal ranks[J].Linear Algebra and Its Applications,2000,321(1/2/3):327-345.
    [8]TIAN Y G,LIU Y H.Extremal ranks of some symmetric matrix expressions with applications[J].SIAM Journal on Matrix Analysis and Applications,2006,28(3):890-905.
    [9]连德忠,袁飞.四元数方程AXAH=B厄米特解中的复矩阵极秩[J].厦门大学学报(自然科学版),2014,53(3):305-309.
    [10]连德忠,谢锦山,吴敏丽,等.四元数方程AXB+CYD=E通解中的复矩阵分量极秩[J].复旦学报(自然科学版),2016,55(3):288-297.