Researches On Some Problems Of Quaternion Matrix Algebra

This dissertation focuses on some problems existing in the quaternion matrix al-gebra and its applications. In this dissertation, we study the estimation of the doubledeterminant of quaternion matrices, simultaneous diagonalization of quaternion matrices,algorithm for computing the transition matrices of Jordan canonical form over quaternionskew-field and solving quaternion matrix equations. The dissertation is divided into fivechapters and main contents are as follows:In chapter one, the preface, we introduce a survey to the development of quaternionmatrix algebra and the background of quaternion and quaternion matrices. Meanwhile,main results in this dissertation are summarized.In chapter two, we study the estimation of the double determinant of the sum of twoquaternion matrices. Firstly, an estimation of the upper bound is given for the double de-terminant of the sum of two arbitrary quaternion matrices by constructive way. Secondly,the lower bound on the double determinant is established especially for the sum of twoquaternion matrices which form an assortive pair. Note that a matrix over number field isalso a quaternion matrix, hence a series related inequalities over the number fields can beestablished. Finally, as applications, some known results are obtained as corollaries and aquestion in the matrix determinant theory is answered completely.Chapter three is divided into two sections. In the first section, the definition of simul-taneous real diagonalization of a pair of quaternion matrices is founded, some necessaryand suficient conditions are discussed for two quaternion matrices can be simultaneouslyreal diagonalized, and an algorithm for computing the simultaneous real diagonalizationof two quaternion matrices is provided. Finally, the simultaneous real diagonalization ofa pair of quaternion matrices is applied in solving quaternion matrix equations. In thesecond section, the definition of simultaneous complex diagonalization of a pair of quater-nion matrices is established, and necessary and suficient conditions of two quaternionmatrices can be simultaneously complex diagonalized are obtained. Moreover, a series ofnecessary and suficient conditions are established for the set of quaternion rectangularmatrices which can be simultaneously complex diagonalized.In chapter four, by the definition of Jordan chains over quaternion skew-field, and bythe corresponding relationship between quaternion matrices and their complex derived ma-trices, a complete algorithm for computing the transition matrices of the Jordan canonical forms of quaternion matrices is given.Note that, since the formulae of Kronecker product over number fields hold no longerfor quaternion matrices, it is often not convenient to handle the quaternion matrix equa-tions cannot be studied by the usual way. In the last chapter, firstly, we study one kindof generalized Sylvester equations, AX fi XB = 0 , and give out the concrete expressionof the general solutions. Secondly, since complex matrix equations are also quaternionmatrix equations, we study the complex matrix equation AX - XB = C . By the spe-cial properties of quaternion matrices, necessary and suficient conditions are obtained forthe matrix equation being consistent. Furthermore, a complete algorithm for solving thisequation is given. Finally, we study one kind of inverse problems and its least-squaresproblems, which improve the existing results also.