|本期目录/Table of Contents|

[1]连德忠*,谢锦山.四元数矩阵方程AXB=C通解中的复矩阵分量极秩[J].厦门大学学报(自然科学版),2019,58(04):543-546.[doi:10.6043/j.issn.0438-0479.201804027]
 LIAN Dezhong*,XIE Jinshan.Extremal ranks of complex components in general solutions of the matric equation AXB=C over quaternion field[J].Journal of Xiamen University(Natural Science),2019,58(04):543-546.[doi:10.6043/j.issn.0438-0479.201804027]
点击复制

四元数矩阵方程AXB=C通解中的复矩阵分量极秩(PDF)
分享到:

《厦门大学学报(自然科学版)》[ISSN:0438-0479/CN:35-1070/N]

卷:
58卷
期数:
2019年04期
页码:
543-546
栏目:
研究论文
出版日期:
2019-07-28

文章信息/Info

Title:
Extremal ranks of complex components in general solutions of the matric equation AXB=C over quaternion field
文章编号:
0438-0479(2019)04-0543-04
作者:
连德忠*谢锦山
龙岩学院数学与信息工程学院,福建 龙岩 364012
Author(s):
LIAN Dezhong*XIE Jinshan
School of Mathematical and Information Engineering,Longyan University,Longyan 364012,China
关键词:
四元数 矩阵方程 复表示 分块矩阵 极秩
Keywords:
quaternion matric equations complex representation block matrix extremal ranks
分类号:
O 151.23
DOI:
10.6043/j.issn.0438-0479.201804027
文献标志码:
A
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AXB=C转换为复数域上的等价复矩阵方程Φ(A)XΦ(B)=Φ(C).同时,利用该复矩阵方程的通解和分块矩阵的极秩性质,求出原四元数矩阵方程通解中复矩阵分量集{X0}和{X1}的最大秩、最小秩公式.作为这些极秩公式的应用,推导出了该四元数矩阵方程通解中包含复矩阵解或全为复矩阵解的充要条件.
Abstract:
By using a complex representation of quaternion matrix Φ(·),the linear matrix equation AXB=C over the quaternion field is changed into the matrix equationΦ(A)X Φ(B)=Φ(C)over the complex field.Then according to general solutions of this complex matrix equation and numerous properties regarding extreme ranks of block matrix, formulas of extreme ranks of complex matrices {X0},{X1} are established.These complex matrices are complex components of general solutions X=X0+X1j of the quaternion matrix equation.As an application,we give necessary and sufficient conditions for following special cases:there exists at least a complex matrix X in general solutions of the matrix equation; and all general solutions of the matrix equation are complex ones.

参考文献/References:

[1] HUNGERFORD T W.Algebra [M].New York:Spring-Verlag,1980:122-151.
[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.[1] HUNGERFORD T W.Algebra [M].New York:Spring-Verlag,1980:122-151.
[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.

备注/Memo

备注/Memo:
收稿日期:2018-04-29录用日期:2019-03-08
基金项目:国家自然科学基金(11601214,11526107); 福建省自然科学基金(2015J05010); 福建省教育厅课程思政项目(KC18084); 福建省教育厅中青年项目(JAT160492,JAT160490)
*通信作者:liandezhong@163.com
更新日期/Last Update: 1900-01-01