Researches On General Solutions To Some Generallized Sylvester Matrix Equations Over The Quaternion Algebra With Applications

In this dissertation,we investigate solvability conditions to some generalized Sylvester matrix equations(systems)over the quaternion algebra with applica-tions.By using generalized inverse and rank of a matrix,we give necessary and sufficient conditions for the existence of the solutions to the matrix equa-tion A1X1B1+A2X2B2 + A3X3B3 = C and the system of matrix equations A1X1 = C1,AX1B1+X2B2 = C3,A2X2 + A3X3B= C2,X3B3 = C4 as well as present expressions of general solutions to these equation and system when the solvability conditions are satisfied.Using the results,we derive solvability con-ditions and expressions of the general solutions to system of quaternion matrix equations A1XB1 = C1,A2XB2 = C2,A3XB3+A4X1B4 + A5X2+ X3B5 = C3 and system of A1XB1=C1,A2XB2=C2,A3XB3=C3,A4X = C4,XB5= C5,and give some new results of exiting literature related to this dissertation.As further applications,we produce necessary and sufficient conditions for the ex-istences and expressions of reducible solutions to the classical matrix equation AXB =C and the system of matrix equations AX = C,XB = D.We also give the necessary and sufficient conditions for two associated networks to have the same branch current and branch voltage and give expressions of the same branch current and branch voltage when the conditions are met.Besides,we in this dissertation discuss the following special problems on Sylvester matrix equa-tion.First,we give necessary and sufficient conditions for the normal Sylvester matrix equation A1X + XB2 = C and its general form A1XB1+A2XB2=C to have solutions as well as present the expressions of the general solution to the two equations when the coefficient matrices axe Hermitian idempotent matrices,respectively.Then,the necessary and sufficient conditions for the existence of complex solutions to the quaternion matrix equation AX+YB=C and the rep-resentations of such solutions are derived when the consistency conditions hold by using two-conjugate of a quaternion matrix.Finally,we establish solvability conditions of row and column conjugated symmetric solutions to the system of complex matrix equations AX = C,XB = D as well as the representations of such solutions when the consistency conditions are met,respectively.As an appli-cation,a set of right eigenvectors belonging to a right eigenvalue of a quaternion matrix is given.Furthermore,some numerical examples are presented to illustrate the results of this paper.