An Analog of the Adjugate Matrix for the Outer Inverse AT,S(2)

Abstract

We investigate the determinantal representation by exploiting the limiting expression for the generalized inverseAT,S(2). We show the equivalent relationship between the existence and limiting expression ofAT,S(2)and some limiting processes of matrices and deduce the new determinantal representations ofAT,S(2), based on some analog of the classical adjoint matrix. Using the analog of the classical adjoint matrix, we present Cramer rules for the restricted matrix equationAXB=D, R(X)⊂T, N(X)⊃S∼.

Complex ZNN for computing time-varying weighted pseudo-inverses

“…Proposition 2. ( [17,18,41]) Let A ∈ C m×n be of rank r, let T be a subspace of C n of dimension s ≤ r, and let S be a subspace of C m of dimension m − s. In addition, suppose that G ∈ C n×m satisfies R(G) = T and N (G) = S. In the case of existence, A (2) T,S can be derived using the following limit representation:…”

“…The starting point in defining the ZNN models for computing the WIPI arises from limit representations (17), (18) and two fundamental properties of A(t) M,N introduced in Lemma 3.…”

“…According to the limit representations (9) and (18), it is reasonable to propose the following two dual fundamental error-monitoring ZFs which represent the basis for ZNN models appropriate for numerical computation of…”

Explicit formulas for determinantal representations of the Drazin inverse solutions of some matrix and differential matrix equations

“…The paper is based on principles used in [23], where we obtained analogs of the Cramer rule for the minimum norm least squares solutions of the matrix equations (1), (2) and (3). Liu et al in [24] deduce the new determinantal representations of A (2) T,S and the Cramer rule for the restricted matrix equation (3) based on these principles as well. Since the Drazin inverse and the group inverse A are outer inverses A (2) T,S for some specific choice of T and S, then the results obtained in [24] generalize in some ways some results of the paper.…”

“…Liu et al in [24] deduce the new determinantal representations of A (2) T,S and the Cramer rule for the restricted matrix equation (3) based on these principles as well. Since the Drazin inverse and the group inverse A are outer inverses A (2) T,S for some specific choice of T and S, then the results obtained in [24] generalize in some ways some results of the paper. But we get the more detailed representation of the Drazin inverse solutions, and therefore we can used their for determinantal representations of solutions of some differential matrix equations.…”