Abstract
The generalized inverse has numerous important applications in aspects of the theoretic research of matrices and statistics. One of the core problems of generalized inverse is finding the necessary and sufficient conditions for the reverse (or the forward) order laws for the generalized inverse of matrix products. In this paper, by using the extremal ranks of the generalized Schur complement, some necessary and sufficient conditions are given for the forward order law for A1{1,2}A2{1,2}…An{1,2}⊆(A1A2…An){1,2}.
Highlights
Throughout this paper, all matrices will be over the complex number field C
Cm×n and Cm denote the set of m × n complex matrices and m-dimensional complex vectors, respectively
The identity matrix of order k is denoted by Ik, and the m × n matrix of all zero entries is denoted by Om×n
Summary
Throughout this paper, all matrices will be over the complex number field C. One of the core problems with the LS above is identifying the conditions under which the following reverse order laws hold: An(1,j,... Another core problem with the LS above is identifying the conditions under which the following forward order laws hold: A1(1,j,... The reverse order laws for the generalized inverse of multiple matrix products (1) yield a class of interesting problems that are fundamental in the theory of the generalized inverse of matrices; see [1,4–6].
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