关于四元数矩阵方程组AX=B,XC=D解的研究
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摘要
本文在四元数体上研究矩阵方程组AX=B,XC=D的(P,Q)-(斜)对称解、反射亚(半)正定解和反射次亚(半)正定解.
全文共分为四章.
第一章主要介绍四元数、四元数矩阵的一些基础知识;各种对称矩阵、亚(半)正定矩阵、次亚(半)正定矩阵的定义和基本性质以及本文所用到的研究矩阵方程的重要工具:广义逆,矩阵表达式的秩等式和双矩阵的同时分解定理.
第二章主要研究以下问题;
问题1:给定A∈H~(s×m),B∈H~(s×n),C∈H~(n×t)和D∈H~(m×t),求X∈H_r~(m×n)(P,Q)或X∈H_a~(m×n)(P,Q)满足矩阵方程组AX=B,XC=D.
问题2:求矩阵方程组AX=B,XC=D的(P,Q)-对称和(P,Q)-斜对称解的最大秩和最小秩及其对应的解.
问题3:当P,Q是厄米对合矩阵时,如果问题1的解集Ω非空,给定E∈H~(m×n),求X_0∈Ω使
‖X_0-E‖=(?)‖X-E‖.
我们通过讨论(P,Q)-(斜)对称矩阵的结构性质,利用矩阵技巧和第一章所给的矩阵理论解决了上述问题,并给出了数值例子.
第三章,通过讨论分块矩阵是亚(半)正定矩阵的充要条件以及反射亚(半)正定矩阵的结构性质,利用双矩阵同时分解定理得到矩阵方程组AX=B,XC=D有亚(半)正定解、反射亚(半)正定解的充要条件,以及矩阵方程组AX=B,XC=D有亚(半)正定解、反射亚(半)正定解时通解的表达式.
第四章,通过讨论分块矩阵是次亚(半)正定矩阵的充要条件以及反射次亚(半)正定矩阵的性质,利用双矩阵同时分解定理得到矩阵方程组AX=B,XC=D有次亚(半)正定解、反射次亚(半)正定解的充要条件,以及矩阵方程组AX=B,XC=D有次亚(半)正定解、反射次亚(半)正定解时通解的表达式.
In this dissertation,we investigate the(P,Q)-(skew)symmetric solution,reflexive re-positive(semi)definite solution and reflexive re-positive per-(semi)definite solution to the system of quaternion matrix equations AX=B,CX=D.
The dissertation is divided into 4 chapters.
In Chapter 1,we introduce the research background and progresses as well as the work we have derived in this dissertation.Some preliminary knowledge used in this paper are also presented,e.g.,the basic matrix theory,symmetric matrix,the generalizcd inverses of matrices,some rank equalities of some matrix expressions, etc.
In Chapter 2,we mainly investigate the following three problems:
Problem 1 Give necessary and sufficient conditions for the existence of and the expressions for the(P,Q)-symmetric and(P,Q)-skewsymmetric solutions to the system of quaternion matrix equations AX=B,XC=D.
Problem 2 Find out the formulas of maximal and minimal ranks of(P,Q)-symmetric and(P,Q)-skewsymmetric solutions to the system.Derive the expressions of the(P,Q)-symmetric and(P,Q)-skewsymmetric solutions with maximal and minimal ranks to the system mentioned above.
Problem 3 IfΩ=φwhereΩis the set of all solutions of Problem 1,given E∈H~(m×n),find X_0∈Ωsuch that
‖X_0-E‖=(?)‖X-E‖where P and Q are all Hermitian involutions.
By using the matrix techniques and matrix theory constructed in chapter 1, we first give a practical method to represent an involutory quaternion matrix and establish a representation for a(P,Q)-symmetric(or(P,Q)-skewsymmetric) matrix. Then,we discuss Problem 1,i.e.,establish necessary and sufficient conditions for the existence of and expressions for(P,Q)-symmetric and(P,Q)-skewsymmetric solutions to the system.Next,we give formulas of extremal ranks of(P,Q)-symmetric and(P,Q)-skewsymmetric solutions to the system and present the(P,Q)-(skew)symmetric solution with extremal ranks to the system.After that,we investigate Problem 3,and give the expression of its solution.We also present a numerical example to illustrate the results.
In Chapter 3,we first give a criterion for a partitioned quaternion matrix to be re-positive(semi)definite,then present a criterion for a quaternion matrix to be reflexive re-positive(semi)definite.Next,we establish a necessary and sufficient condition for the existence of re-positive(semi)definite solution to the system of quaternion matrix equations AX=B,XC=D as well as an expression of the general solution.Based on these results,we establish a necessary and sufficient condition for the existence of and an expression for reflexive re-positive(semi)definite solution to the system mentioned above.
In Chapter 4,we consider the reflexive re-positive per-(semi)definite solution to the system of quaternion matrix equations AX=B,CX=D.In this Chapter, we derive a necessary and sufficient condition for the existence of and an expression for the reflexive re-positive per-(semi)definite solution to the system of quaternion matrix equations AX=B,XC=D.In addition,we give a necessary and sufficient condition for the existencc of and an expression for the re-postive per-(semi)definite solution to the system.The criteria for a partitioned quaternion matrix to be re-positive per-(semi)definite and a quaternion matrix to be reflexive re-positive per-(semi) definite are also established.
全文共分为四章.
第一章主要介绍四元数、四元数矩阵的一些基础知识;各种对称矩阵、亚(半)正定矩阵、次亚(半)正定矩阵的定义和基本性质以及本文所用到的研究矩阵方程的重要工具:广义逆,矩阵表达式的秩等式和双矩阵的同时分解定理.
第二章主要研究以下问题;
问题1:给定A∈H~(s×m),B∈H~(s×n),C∈H~(n×t)和D∈H~(m×t),求X∈H_r~(m×n)(P,Q)或X∈H_a~(m×n)(P,Q)满足矩阵方程组AX=B,XC=D.
问题2:求矩阵方程组AX=B,XC=D的(P,Q)-对称和(P,Q)-斜对称解的最大秩和最小秩及其对应的解.
问题3:当P,Q是厄米对合矩阵时,如果问题1的解集Ω非空,给定E∈H~(m×n),求X_0∈Ω使
‖X_0-E‖=(?)‖X-E‖.
我们通过讨论(P,Q)-(斜)对称矩阵的结构性质,利用矩阵技巧和第一章所给的矩阵理论解决了上述问题,并给出了数值例子.
第三章,通过讨论分块矩阵是亚(半)正定矩阵的充要条件以及反射亚(半)正定矩阵的结构性质,利用双矩阵同时分解定理得到矩阵方程组AX=B,XC=D有亚(半)正定解、反射亚(半)正定解的充要条件,以及矩阵方程组AX=B,XC=D有亚(半)正定解、反射亚(半)正定解时通解的表达式.
第四章,通过讨论分块矩阵是次亚(半)正定矩阵的充要条件以及反射次亚(半)正定矩阵的性质,利用双矩阵同时分解定理得到矩阵方程组AX=B,XC=D有次亚(半)正定解、反射次亚(半)正定解的充要条件,以及矩阵方程组AX=B,XC=D有次亚(半)正定解、反射次亚(半)正定解时通解的表达式.
In this dissertation,we investigate the(P,Q)-(skew)symmetric solution,reflexive re-positive(semi)definite solution and reflexive re-positive per-(semi)definite solution to the system of quaternion matrix equations AX=B,CX=D.
The dissertation is divided into 4 chapters.
In Chapter 1,we introduce the research background and progresses as well as the work we have derived in this dissertation.Some preliminary knowledge used in this paper are also presented,e.g.,the basic matrix theory,symmetric matrix,the generalizcd inverses of matrices,some rank equalities of some matrix expressions, etc.
In Chapter 2,we mainly investigate the following three problems:
Problem 1 Give necessary and sufficient conditions for the existence of and the expressions for the(P,Q)-symmetric and(P,Q)-skewsymmetric solutions to the system of quaternion matrix equations AX=B,XC=D.
Problem 2 Find out the formulas of maximal and minimal ranks of(P,Q)-symmetric and(P,Q)-skewsymmetric solutions to the system.Derive the expressions of the(P,Q)-symmetric and(P,Q)-skewsymmetric solutions with maximal and minimal ranks to the system mentioned above.
Problem 3 IfΩ=φwhereΩis the set of all solutions of Problem 1,given E∈H~(m×n),find X_0∈Ωsuch that
‖X_0-E‖=(?)‖X-E‖where P and Q are all Hermitian involutions.
By using the matrix techniques and matrix theory constructed in chapter 1, we first give a practical method to represent an involutory quaternion matrix and establish a representation for a(P,Q)-symmetric(or(P,Q)-skewsymmetric) matrix. Then,we discuss Problem 1,i.e.,establish necessary and sufficient conditions for the existence of and expressions for(P,Q)-symmetric and(P,Q)-skewsymmetric solutions to the system.Next,we give formulas of extremal ranks of(P,Q)-symmetric and(P,Q)-skewsymmetric solutions to the system and present the(P,Q)-(skew)symmetric solution with extremal ranks to the system.After that,we investigate Problem 3,and give the expression of its solution.We also present a numerical example to illustrate the results.
In Chapter 3,we first give a criterion for a partitioned quaternion matrix to be re-positive(semi)definite,then present a criterion for a quaternion matrix to be reflexive re-positive(semi)definite.Next,we establish a necessary and sufficient condition for the existence of re-positive(semi)definite solution to the system of quaternion matrix equations AX=B,XC=D as well as an expression of the general solution.Based on these results,we establish a necessary and sufficient condition for the existence of and an expression for reflexive re-positive(semi)definite solution to the system mentioned above.
In Chapter 4,we consider the reflexive re-positive per-(semi)definite solution to the system of quaternion matrix equations AX=B,CX=D.In this Chapter, we derive a necessary and sufficient condition for the existence of and an expression for the reflexive re-positive per-(semi)definite solution to the system of quaternion matrix equations AX=B,XC=D.In addition,we give a necessary and sufficient condition for the existencc of and an expression for the re-postive per-(semi)definite solution to the system.The criteria for a partitioned quaternion matrix to be re-positive per-(semi)definite and a quaternion matrix to be reflexive re-positive per-(semi) definite are also established.
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