Some Investigations On The Solutions To Some Systems Of Quaternions Matrix Equations

Matrix equation (group) is a very important branch of the matrix theory.It plays a very important role in the linear matrix equation (group) research,and has a wide range of applications in mechanics, cybernetics,remote sensing control.The quaternion matrix do not exchange,so it’s challenging to research the quaternion matrix equation (group).In recent forty years,the research of matrix equation is in the ascendant.It has an important significance in theory,and has a wide range of applications in mechanics,cybernetics,theoretical physics,theoretical electrical technology,remote sensing technology and so on.Considering that the quaternion and quaternion matrix have a wide range of applications,the research of the quaternion matrix equation theory and its numerical calculation is particularly important. First of all,appling operational methods of generalized inverse and ranks of matrix,the sufficient and necessary conditions for the existence of the solutions over the quaternion linear matrix equations and the expression form of solutions are investigated,so the extreme ranks of the quaternion matrix equation expression under the certain conditions are deduced.Secondly,as the application of the above conclusions,the sufficient and necessary conditions for the existence of symmetric and skewsymmetric solutions over the quaternion matrix equations and the expression form of solutions when the solutions exist are discussed.Finally,using the singular value decomposition of quaternion self-conjugate matrix,the necessary and sufficient conditions for a kind of quaternion matrix that have general solution,the self-conjugate solution and the positive definite self-conjugate solution and the expression forms of general solutions are given.Firstly, in this paper,the sufficient and necessary conditions for the existence of solutions over matrix equations A1X1=C1, A2X1=C2, A3X2=C3, A4X2=C4, A5X1B1+A6X2B2=C5are discussed.Moreover,the expressions of general solutions and rank solutions when the solutions exist,the extreme ranks of general solutions and the minimum norm are presented.Secondly,the extreme ranks of the linear matrix expression f(X1,X2)=C5-A5X1B1-A6X2B2in condition of the system of linear equations A1X1=C1, A2X1=C2, A3X2=C3, A4X2=C4are deduced. Among them,C5,A5,B1,B1,A6,B2are the known matrix,X1,X2are unknown matrix.Thirdly,as the application of the above conclusions, the sufficient and necessary conditions for the existence of (P,Q)-symmetric solution,(P,Q)-skewsymmetric solution, P-symmetricand solution and P-skewsymmetric solution over the system of equations AaX=Ca, AbX=Cb, AcXBa=C5and the expressions when the solutions exist are discussed.It is concluded that the literature in this paper is the conclusion of the special situation.Fourth, using the singular value decomposition of quaternion self-conjugate matrix,the necessary and sufficient conditions for quaternion matrix equation AX+YB=C that have general solution,the self-conjugate solution and the positive definite self-conjugate solution, and the expression forms of the general solutions are discussed.