On Properties Of Quaternion Matrix And Quaternion Matrix Equation(s)

The quaternion matrix and the quaternion matrix equation(s) have been widely applied in Geostatics,Quantum mechanics,Automation,Robot technology and so on.Because of these,the quaternion matrix and the quaternion matrix equation(s) are always one of the main research topics in both pure and applied research fields.In this paper,we mainly discuss the properties of self-conjugate quaternion matrices and the existence of solutions of the quaternion matrix equation(s).By using the singular value decomposition(SVD) and the generalized inverse of matrices, we obtain the following results.1.A necessary and sufficient condition for the existence of the general anti-symmetric unitary anti-symmetric solutions of the quaternion matrix equation(XA₁, XA₂)=(B₁,B₂) is obtained,and the general form of the solutions is given.2.A necessary and sufficient condition for the existence of the minor(inverse) self-conjugate and skewpositive(semi-)definite solutions to the system of matrix equations over the quaternion field(XA,XB)=(A,O) is obtained,and the general form of the solutions is given.3.A necessary and sufficient condition for the existence of the self-conjugate positive semi-definite and semi-positive subdefinite solutions of the quaternion matrix equation(A^*XA,B^*XB)=(C,D) is obtained,and the general form of the solutions is given.Finally,two necessary and sufficient condutions are preseated,which are related to the trace inequalities of self-conjugate quaternion matrices.