四元数矩阵方程实解与复解的研究
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摘要
本文在四元数除环上研究了若干矩阵方程组一般解的最大与最小秩,并由此导出了某些四元数矩阵方程组有实解和复解的充要条件以及实解和复解的表达式。这些结果进一步丰富和发展了四元数矩阵代数。
全文共分为五章,第一章介绍了四元数、四元数矩阵、四元数矩阵方程、矩阵的秩以及矩阵的广义逆的一些研究背景、研究进展以及本文所做的工作。另外还给出了本文要用到的一些预备知识。第二章研究四元数矩阵表达式C_4-A_4XB_4在四元数矩阵方程组A_1X=C_1,XB_2=C_2,A_3XB_3=C_3有解的条件下的最大秩最小秩,并利用秩的等式给出了四元数矩阵方程组A_1X=C_1,XB_2=C_2,A_3XB_3=C_3,A_4XB_4=C_4有一般解的充要条件。第三章给出了四元数矩阵方程AXB=C一般解的最大最小秩,并由此导出了它有实解和复解的充要条件及解的表达式。作为应用,还给出了四元数矩阵方程组A_1XB_1=C_1,A_2XB_2=C_2有实解和复解的充要条件。第四章给出了四元数矩阵方程组A_1XB_1=C_1,A_2XB_2=C_2一般解的最大最小秩,并由此导出了它有实解和复解的充要条件及解的表达式。作为应用,还给出了四元数矩阵方程组A_1XB_1=C_1,A_2XB_2=C_2,A_3XB_3=C_3有实解和复解的充要条件。第五章给出了四元数矩阵方程组A_1X=C_1,XB_2=C_2,A_3XB_3=C_3一般解的最大最小秩,并由此导出了它有实解和复解的充要条件及解的表达式。作为应用,还给出了四元数矩阵方程组A_1X=C_1,XB_2=C_2,A_3XB_3=C_3,A_4XB_4=C_4有实解和复解的充要条件。
In this dissertation,by studying the maximal and minimal ranks of the general solutions to certain systems of quaternion matrix equations,necessary and sufficient conditions for the existences and expressions of real and complex solutions to some systems of quaternion matrix equations are given.These results further enrich and develop the quaternion matrix algebra.
The dissertation is divided into 5 chapters.In Chapter 1,we introduce the research background and progresses of quaternion,quaternion matrices,quaternion matrix equations,extremal ranks of matrix expressions,generalized inverse of matrix as well as the work we have done in this dissertation.Some preliminary knowledge used in this paper are also presented.In Chapter 2,we give the maximal and minimal ranks of the quaternion matrix expression C_4 -- A_4XB_4 subject to quaternion matrix equations A_1X = C_1,XB_2 = C_2,A_3XB_3 = C_3, ncccssary and sufficient conditions for the solvability of the quaternion matrix equations A_1X = C_1,XB_2 = C_2,A_3XB_3 = C_3,A_4XB_4 = C_4 are derived by rank equalities.In Chapter 3,we give the maximal and minimal ranks of the general solution to quaternion matrix equation AXB = C,and derive necessary and sufficient conditions for the existences and the expressions of real and complex solutions to the matrix equation.As an application,necessary and sufficient conditions for quaternion matrix equations A_1XB_1 = C_1,A_2XB_2 = C_2 to have real and complex solutions are presented.In Chapter 4,we investigate the maximal and minimal ranks of the general solution to quaternion matrix equations A_1XB_1 = C_1,A_2XB_2 = C_2,and derive necessary and sufficient conditions for the existences and the expressions of real and complex solutions to the matrix equations.As an application,necessary and sufficient conditions for quaternion matrix equations A_1XB_1 = C_1,A_2XB_2 = C_2,A_3XB_3 = C_3 to have real and complex solutions are given.In the last Chapter,we investi- gate the extremal ranks of the general solution to quaternion matrix equations A_1X = C_1,XB_2 = C_2,A_3XB_3 = C_3,and derive necessary and sufficient conditions for the existences and the expressions of real and complex solutions to the matrix equations.As an application,necessary and sufficient conditions for quaternion matrix equations A_1X = C_1,XB_2 = C_2,A_3XB_3 = C_3,A_4XB_4 = C_4 to have real and complex solutions are established.
全文共分为五章,第一章介绍了四元数、四元数矩阵、四元数矩阵方程、矩阵的秩以及矩阵的广义逆的一些研究背景、研究进展以及本文所做的工作。另外还给出了本文要用到的一些预备知识。第二章研究四元数矩阵表达式C_4-A_4XB_4在四元数矩阵方程组A_1X=C_1,XB_2=C_2,A_3XB_3=C_3有解的条件下的最大秩最小秩,并利用秩的等式给出了四元数矩阵方程组A_1X=C_1,XB_2=C_2,A_3XB_3=C_3,A_4XB_4=C_4有一般解的充要条件。第三章给出了四元数矩阵方程AXB=C一般解的最大最小秩,并由此导出了它有实解和复解的充要条件及解的表达式。作为应用,还给出了四元数矩阵方程组A_1XB_1=C_1,A_2XB_2=C_2有实解和复解的充要条件。第四章给出了四元数矩阵方程组A_1XB_1=C_1,A_2XB_2=C_2一般解的最大最小秩,并由此导出了它有实解和复解的充要条件及解的表达式。作为应用,还给出了四元数矩阵方程组A_1XB_1=C_1,A_2XB_2=C_2,A_3XB_3=C_3有实解和复解的充要条件。第五章给出了四元数矩阵方程组A_1X=C_1,XB_2=C_2,A_3XB_3=C_3一般解的最大最小秩,并由此导出了它有实解和复解的充要条件及解的表达式。作为应用,还给出了四元数矩阵方程组A_1X=C_1,XB_2=C_2,A_3XB_3=C_3,A_4XB_4=C_4有实解和复解的充要条件。
In this dissertation,by studying the maximal and minimal ranks of the general solutions to certain systems of quaternion matrix equations,necessary and sufficient conditions for the existences and expressions of real and complex solutions to some systems of quaternion matrix equations are given.These results further enrich and develop the quaternion matrix algebra.
The dissertation is divided into 5 chapters.In Chapter 1,we introduce the research background and progresses of quaternion,quaternion matrices,quaternion matrix equations,extremal ranks of matrix expressions,generalized inverse of matrix as well as the work we have done in this dissertation.Some preliminary knowledge used in this paper are also presented.In Chapter 2,we give the maximal and minimal ranks of the quaternion matrix expression C_4 -- A_4XB_4 subject to quaternion matrix equations A_1X = C_1,XB_2 = C_2,A_3XB_3 = C_3, ncccssary and sufficient conditions for the solvability of the quaternion matrix equations A_1X = C_1,XB_2 = C_2,A_3XB_3 = C_3,A_4XB_4 = C_4 are derived by rank equalities.In Chapter 3,we give the maximal and minimal ranks of the general solution to quaternion matrix equation AXB = C,and derive necessary and sufficient conditions for the existences and the expressions of real and complex solutions to the matrix equation.As an application,necessary and sufficient conditions for quaternion matrix equations A_1XB_1 = C_1,A_2XB_2 = C_2 to have real and complex solutions are presented.In Chapter 4,we investigate the maximal and minimal ranks of the general solution to quaternion matrix equations A_1XB_1 = C_1,A_2XB_2 = C_2,and derive necessary and sufficient conditions for the existences and the expressions of real and complex solutions to the matrix equations.As an application,necessary and sufficient conditions for quaternion matrix equations A_1XB_1 = C_1,A_2XB_2 = C_2,A_3XB_3 = C_3 to have real and complex solutions are given.In the last Chapter,we investi- gate the extremal ranks of the general solution to quaternion matrix equations A_1X = C_1,XB_2 = C_2,A_3XB_3 = C_3,and derive necessary and sufficient conditions for the existences and the expressions of real and complex solutions to the matrix equations.As an application,necessary and sufficient conditions for quaternion matrix equations A_1X = C_1,XB_2 = C_2,A_3XB_3 = C_3,A_4XB_4 = C_4 to have real and complex solutions are established.
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