Abstract
We classify, extend and unify various generalizations of weighted Moore-Penrose inverses in indefinite inner product spaces. New kinds of generalized inverses are introduced for this purpose. These generalized inverses are included in the more general class called as the weighted indefinite pseudoinverses (WIPI), which represents an extension of the Minkowski inverse (MI), the weighted Minkowski inverse (WMI), and the generalized weighted Moore- Penrose (GWM-P) inverse. The WIPI generalized inverses are introduced on the basis of two Hermitian invertible matrices and two Hermitian involuntary matrices and represented as particular outer inverses with prescribed ranges and null spaces, in terms of appropriate full-rank and limiting representations. Application of introduced generalized inverses in solving some indefinite least squares problems is considered. New Zeroing Neural Network (ZNN) models for computing the WIPI are developed using derived full-rank and limiting representations. The convergence behavior of the proposed ZNN models is investigated. Numerical simulation results are presented.
Highlights
The indefinite inner product associated with an invertible Hermitian matrixJ is defined by u, v J = (u, Jv) = u∗Jv, where (x, y) = x∗y denotes the conventional inner product in a Hilbert space
(1) The weighted indefinite pseudoinverses (WIPI), WPPI and weighted Minkowski inverse (WMI) are represented as a particular outer inverse with prescribed range and null space
We investigate generalizations of Penrose equations in indefinite inner product spaces
Summary
Weighted Minkowski inverse; indefinite inner product; Zhang neural network; outer inverse; time-varying complex matrix. The dynamical equation and corresponding artificial recurrent neural network for computing the Drazin inverse of an arbitrary square real matrix, without any restriction on eigenvalues of its rank invariant powers, were proposed in [29]. Various ZNN models for computing online time-varying Moore-Penrose inverse of a full-rank matrix were proposed in [50]. ZNN models for computing outer inverses with prescribed range and null space of time-varying complex matrix were presented recently in [38]. Provided ZNN models will be applicable in the computation of the generalized inverses WPPI, PPI, WMI and MI The dynamics of these neural networks are based on matrixvalued ZFs arising from the limit representations of the WIPI.
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