若干四元数矩阵方程解的秩及其应用
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摘要
本文我们主要运用矩阵广义逆和矩阵秩的方法,去研究四元数体上的若干具有重要意义的矩阵方程和方程组解中实矩阵的表达式及其极秩,并且利用极秩的性质给出了矩阵方程和方程组有特殊解的充分必要条件,以及一些相关的应用.
全文共分为四章:
第一章主要介绍了四元数、四元数矩阵的基本知识和基本性质以及四元数矩阵方程的研究背景、研究现状和本文所做的工作,另外还有一些重要的矩阵秩的公式.
第二章我们在四元数体上研究了矩阵方程AXB+CYD=E,方程组A_1XB_1 + C_1YD_1 = E_1,A_2XB_2 + C_2YD_2 = E_2以及广义Sylvester方程组A_1X+YB_1 = C_1, A_2X+YB_2 = C_2.给出了方程(组)的解中实矩阵的具体表达式,并推导出了每个实矩阵的最秩,最后用秩等式给出了方程有特殊解的充要条件.
第三章我们利用矩阵的技巧对四元数体上矩阵方程AXB+CYD=E和方程组A_1XB_1 = C_1, A_2XB_2= C_2的解做矩阵分块,再运用求矩阵秩的方法推导出解中各个分块的极秩,最后给出了方程(组)有各种特殊分块解的充要条件以及一些相关应用.
第四章主要研究了四元数体上矩阵的Moore-Penrose逆、Drazin逆、群逆和{1,3),{1,4)-逆中各个实矩阵的秩等式和一些重要的秩的不等式,以及矩阵的各种广义逆为实矩阵或者是纯虚数矩阵的充要条件.
In this dissertation, by using the fundamental theory and methods of generalized inverse and rank of matrix, we investigate solutions of some classic linear equation and systems of linear matrix equations over quaternion division algebra. The expressions and the extreme ranks of real matrix in the solutions of matrix equations mentioned above are discussed. Moreover, the necessary and sufficient conditions for the existence of some special solutions over quaternion division algebra are presented. These results not only further enrich and develop the quaternion matrix algebra, but also very useful in appliced.
The main contents are as the follows:
In Chaper 1, we introduce the research background and progresses of quaternion, quaterion matrices and quaternion matrix equations as well as the work have been done in this thesis. Some preliminary knowledge of extreme rank formulas are also presented.
Based on the preliminary knowledge of extreme rank formulas, in Chaper 2, we consider the linear equation AXB + CYD = E, the system of linear matrix equations A_1XB_1 + C_1YD_1 = E_1,A_2XB_2 + C_2YD_2 = E_2, and the system of Sylvester matrix equations A_1X+YB_1 = C_1, A_2X+YB_2 = C_2 over quaternion division algebra. The expressions of real matrices in solutions to the equations mentioned above are given. The extreme rank of real matrices in solutions to the equations mentioned above are derivated. As applications, we established the necessary and sufficient conditions for the existence of some special solutions of above systems over quaternion division algebra.
In Chaper 3, the necessary and sufficient conditions for the existence of some special partitioned solutions to the quaternion matrix equation AXB + CYD = E and the system of quaternion matrix equations A_1XB_1 = C_1, A_2XB_2 = C_2 are discribed, which use the techniques of constructing partitioned matrices and extreme rank. Some corresponding results of special cases are also considered.
In the last Chapter, the rank of Moore-Penrose inverse, Drazin inverse and Group inverse are investiged. We also present the maximal and minimal ranks of {1,3}, {1,4}-inverse and establish the ncessary and sufficient condition for the existence of special {1,3}, {1,4}- inverse.
全文共分为四章:
第一章主要介绍了四元数、四元数矩阵的基本知识和基本性质以及四元数矩阵方程的研究背景、研究现状和本文所做的工作,另外还有一些重要的矩阵秩的公式.
第二章我们在四元数体上研究了矩阵方程AXB+CYD=E,方程组A_1XB_1 + C_1YD_1 = E_1,A_2XB_2 + C_2YD_2 = E_2以及广义Sylvester方程组A_1X+YB_1 = C_1, A_2X+YB_2 = C_2.给出了方程(组)的解中实矩阵的具体表达式,并推导出了每个实矩阵的最秩,最后用秩等式给出了方程有特殊解的充要条件.
第三章我们利用矩阵的技巧对四元数体上矩阵方程AXB+CYD=E和方程组A_1XB_1 = C_1, A_2XB_2= C_2的解做矩阵分块,再运用求矩阵秩的方法推导出解中各个分块的极秩,最后给出了方程(组)有各种特殊分块解的充要条件以及一些相关应用.
第四章主要研究了四元数体上矩阵的Moore-Penrose逆、Drazin逆、群逆和{1,3),{1,4)-逆中各个实矩阵的秩等式和一些重要的秩的不等式,以及矩阵的各种广义逆为实矩阵或者是纯虚数矩阵的充要条件.
In this dissertation, by using the fundamental theory and methods of generalized inverse and rank of matrix, we investigate solutions of some classic linear equation and systems of linear matrix equations over quaternion division algebra. The expressions and the extreme ranks of real matrix in the solutions of matrix equations mentioned above are discussed. Moreover, the necessary and sufficient conditions for the existence of some special solutions over quaternion division algebra are presented. These results not only further enrich and develop the quaternion matrix algebra, but also very useful in appliced.
The main contents are as the follows:
In Chaper 1, we introduce the research background and progresses of quaternion, quaterion matrices and quaternion matrix equations as well as the work have been done in this thesis. Some preliminary knowledge of extreme rank formulas are also presented.
Based on the preliminary knowledge of extreme rank formulas, in Chaper 2, we consider the linear equation AXB + CYD = E, the system of linear matrix equations A_1XB_1 + C_1YD_1 = E_1,A_2XB_2 + C_2YD_2 = E_2, and the system of Sylvester matrix equations A_1X+YB_1 = C_1, A_2X+YB_2 = C_2 over quaternion division algebra. The expressions of real matrices in solutions to the equations mentioned above are given. The extreme rank of real matrices in solutions to the equations mentioned above are derivated. As applications, we established the necessary and sufficient conditions for the existence of some special solutions of above systems over quaternion division algebra.
In Chaper 3, the necessary and sufficient conditions for the existence of some special partitioned solutions to the quaternion matrix equation AXB + CYD = E and the system of quaternion matrix equations A_1XB_1 = C_1, A_2XB_2 = C_2 are discribed, which use the techniques of constructing partitioned matrices and extreme rank. Some corresponding results of special cases are also considered.
In the last Chapter, the rank of Moore-Penrose inverse, Drazin inverse and Group inverse are investiged. We also present the maximal and minimal ranks of {1,3}, {1,4}-inverse and establish the ncessary and sufficient condition for the existence of special {1,3}, {1,4}- inverse.
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