Ranks Of Solutions To Some Quaternion Matrix Equations With Applications

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 A₁XB₁ + C₁YD₁ = E₁,A₂XB₂ + C₂YD₂ = E₂, and the system of Sylvester matrix equations A₁X+YB₁ = C₁, A₂X+YB₂ = C₂ 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 A₁XB₁ = C₁, A₂XB₂ = C₂ 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.