Mixed-type Reverse-order Laws And Minkowski Inverse

The theory and methods of generalized inverse are important basic tools in all mathematical disciplines, and have extensive applications in economics, statistics, surveying, optimization techniques, information processing, automatic control, engineering techniques, operations research and so on. Especially, generalized inverse matrices are indispensable studying tools in least square problems, the rectangular or ill-linear problems, the nonlinear problems, the non-constrained or constrained linear programming problems, control and identification of system problems, electronic net problems and so on. In recent years, with the study of theory and computation of generalized inverse matrices, the reverse order laws for the generalized inverses of the multiple matrix products and the theory for the Minkowski inverse in Minkowski space had corresponding development.In this paper, first of all, we study a group of mixed-type reverse order laws for generalized inverses of a triple matrix product by using the expressions of maximal and minimal ranks of generalized Schur complement, then we discuss the Minkowski inverse and its properties.