Recurrent Neural Network for Computing the Drazin Inverse

Abstract

This paper presents a recurrent neural network (RNN) for computing the Drazin inverse of a real matrix in real time. This recurrent neural network (RNN) is composed of n independent parts (subnetworks), where n is the order of the input matrix. These subnetworks can operate concurrently, so parallel and distributed processing can be achieved. In this way, the computational advantages over the existing sequential algorithms can be attained in real-time applications. The RNN defined in this paper is convenient for an implementation in an electronic circuit. The number of neurons in the neural network is the same as the number of elements in the output matrix, which represents the Drazin inverse. The difference between the proposed RNN and the existing ones for the Drazin inverse computation lies in their network architecture and dynamics. The conditions that ensure the stability of the defined RNN as well as its convergence toward the Drazin inverse are considered. In addition, illustrative examples and examples of application to the practical engineering problems are discussed to show the efficacy of the proposed neural network.

Conditions for Existence, Representations, and Computation of Matrix Generalized Inverses

“…A feedforward neural network architecture for computing the Drazin inverse was proposed in [19]. The dynamic equation and induced gradient recurrent neural network for computing the Drazin inverse were defined in [24]. Two gradient-based RNNs for generating outer inverses with prescribed range and null space in the time-invariant case were introduced in [25].…”

Some Results on the Symmetric Representation of the Generalized Drazin Inverse in a Banach Algebra

“…Proof. By Lemma 8, it also leads to (10). By Lemma 3, we can prove that (a + b) d exists if and only if (a 22 + b 2 ) d exists; that is, (a + b) d exists if and only if b π (a + b) is generalized Drazin invertible.…”

“…The generalized inverse in a matrix or operator theory is very useful in scientific calculation and in various engineering technologies [2][3][4]. It is well known that the Drazin inverse has been applied in a few fields, such as statistics and probability [5], ordinary differential equations [6], Markov chains [7], operator matrices [8], neural network models [9,10], and the references therein. In [11], a study of the Drazin inverse for bounded linear operators in a Banach space X is given, when 0 is an isolated spectral point of the operator.…”

ZNN models for computing matrix inverse based on hyperpower iterative methods

“…ZNN models for online time-varying full-rank matrix pseudoinversion were considered in [24]. An RNN with the linear activation function for the Drazin inverse computation was proposed by Stanimirović, Zivković, and Wei in [17]. The relationship between the Zhang matrix inverse and the Drazin inverse, discovered in [25], leads to the same dynamic state equation which was considered in [17] in the time invariant matrix case.…”

“…An RNN with the linear activation function for the Drazin inverse computation was proposed by Stanimirović, Zivković, and Wei in [17]. The relationship between the Zhang matrix inverse and the Drazin inverse, discovered in [25], leads to the same dynamic state equation which was considered in [17] in the time invariant matrix case. 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 [16].…”