Investigations Of The (skew-)Hermitian Solutions To A Quaternion Matrix Equation AXA~* + BYB~* = C

In this dissertation,we investigate the (skew-)Hermitian solutions and thedecomposition of the (skew-)Hermitian solutions to the quaternion matrix equa-tion AXA^*+BYB^*=C. By using the fundamental theory and methods ofgeneralized inverse and rank of matrix, we give the expressions and the extremeranks of the (skew-)Hermitian solutions.These results not only further enrich anddevelop the quaternion matrix algebra, but also very useful in appliced.The dissertation is divided into 4 chapters.In chaper 1, we introduce the research background and progresses as well asthe work we have derived in this dissertation. Some preliminary knowledge usedin this paper are also presented , e.g., the (skew-)Hermitian matrices, the (skew-)Hermitian generalized inverses of matrices and some extreme rank formulas.Chaper 2, We derive the maximal and minimal ranks of (skew-)Hermitiansolutions X, Y to the quaternion matrix equation AXA^*+BYB^*=C, and giveformulas of extreme ranks of real matrices X₁, X₂, X₃, X₄, Y₁, Y₂, Y₃, Y₄ in (skew-)Hermitian solutions X = X₁ + X₂i + X₃j + X4k, Y = Y₁ + Y₂i + Y₃j + Y₄k, aswell as expressions of such solutions. Moreover we established the necessary andsu?cient conditions for the existence of some special solutions of above systemsover quaternion algebra. As applications, some corresponding results of specialcases are also considered.In chaper 3,by using the techniques of constructing partitioned the (skew-)Hermitian solutions X, Y of quaternion matrix equation AXA^*+BYB^*=C intoa 2-by-2 block formas, we derive the maximal and minimal ranks of submatrices inthe (skew-)Hermitian solutions. Moreover the necessary and su?cient conditionsfor the existence of some special partitioned solutions to the quaternion matrixequation are presented respectively. Some known results can be regarded asspecial cases of the results in this paper. At last, we consider the independence of submatrices in (skew-)Hermitian solution to the quaternion matrix equationmentioned above.In chaper 4, we investigate quaternion matrix equations AiXiA?i + BiYiBi? =Ci, i = 1, 2, 3 are consistent, respectively. We give some necessary and su?cientconditions for a (skew-)Hermitian solution of the first equation to decomposethe sum of the (skew-)Hermitian solutions of the other two quaternion matrixequations. Some consequences and applications are obtained for the additivedecompositions. In particular, we present necessary and su?cient conditions forHermitian and skew-Hermitian generalized inverse equalitiesto hold over quaternion algebra respectively.