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Abstract: In this paper, we give some closed-form formulas for calculating maximal and minimal ranks and inertias of with respect to , where is given, is common Hermitian least squares solutions to matrix equations and . As application, we derive necessary and sufficient conditions for in the Löwner partial ordering. In addition, we give identifying conditions for the existence of definite common Hermitian least squares solutions to matrix equations and . [Copyright &y& Elsevier]

Abstract In this paper, we propose an efficient method for special least squares solution of the complex matrix equation (A X B , C X D) = (E , F). By using the real representation matrices of complex matrices, the particular structure of the real representation matrices, the Moore–Penrose generalized inverse and the Kronecker product, we obtain the explicit expression of the minimal norm least squares Hermitian solution of the complex matrix equation (A X B , C X D) = (E , F) , which was studied by a product of matrices and vectors in Wang et al. (2016). Our resulting formulas only involve real matrices, and the corresponding algorithm only performs real arithmetic. Therefore our proposed method is more effective and portable. Finally, we give three numerical examples to illustrate the effectiveness of our proposed method. [ABSTRACT FROM AUTHOR]

In this paper, by using the real representations of quaternion matrices, the particular structure of the real representations of quaternion matrices, the Kronecker product of matrices and the Moore–Penrose generalized inverse, we obtain the expressions of the minimal norm least squares solution, the pure imaginary least squares solution, and the real least squares solution for the quaternion matrix equation A X B + C X D = E , respectively. Our resulting formulas only involve real matrices, and therefore are simpler than those reported in Yuan (2014). The corresponding algorithms only perform real arithmetic which also consider the particular structure of the real representations of quaternion matrices, and therefore are very efficient and portable. Numerical examples are provided to illustrate the efficiency of our algorithms. [ABSTRACT FROM AUTHOR]

Abstract: In this article we solve the following three kinds of problems. First, some formulas for the minimal rank of the submatrices in a solution of matrix equation and the minimal and maximal rank of itself are derived by using the matrix rank method. From these formulas, necessary and sufficient conditions are given for to be nonsingular or the submatrices to be zero, respectively. Second, some formulas for the minimal rank of , and the corresponding expressions of submatrices of are investigated. Combined with matrix decomposition, necessary and sufficient conditions are given for the existence of solutions to be Hermitian, local Hermitian, Skew-Hermitian and local Skew-Hermitian, respectively. Third, necessary and sufficient conditions are given for the existence of solutions to be local positive (negative) semidefinite, and for some Hermitian solution with zero submatrix, the structure form of a positive (negative) semidefinite solution are obtained using results of inertia. [ABSTRACT FROM AUTHOR]