Drazin inverse and M-P inverse have many important applications in the aspects of theoretic research of operator and statistics. In this article, we will exhibit under suitable conditions a neat relationship between the Drazin inverse of A + B and the Drazin inverses of the individual terms A and B . Furthermore, with the same thread, we will give an expression of the M-P inverse of A + B in terms of only the M-P inverses of matrices A and B .
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}.
The M-P inverse has many important applications in the aspects of theoretic research of operator and statistics. One of the core problems in M-P inverse is to find the M-P inverse of operator product. In this paper, by using the technique of matrix form of bounded linear operators, we study the existence and give an expression of the M-P inverse of an operator product. In particular, by applying this expression, we introduce a formula for the M-P inverse of a sum of two operators.
The relationship between the forward order law product A? 1A? 2...A? n of
the Moore-Penrose inverses of A1,A2,... ,An and the seven common types of
generalized inverse of A1A2...An will be studied in this paper.
Especially, we will give the necessary and sufficient condition for the n
terms forward order law (A1A2...An)? = A? 1A? 2...A?n.
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