Drazin Inverse Research Articles

General exact solutions of certain second-order homogeneous algebraic differential equations

The general exact solutions of the second-order homogeneous algebraic differential equation(Section.Display) is studied, where and are three known bounded linear operators on a Banach space . If such that is invertible, an anti-triangular operator matrix is defined. When is Drazin invertible, a necessary and sufficient condition is obtained in terms of and its Drazin inverse under which becomes a solution of (0.1). Specifically, a formula for is derived under the assumptions that is a Hilbert space, and are positive semi-definite operators on with the summation being positive definite.

Characterizations of DMP inverse in a Hilbert space

Let $$\mathcal {H}$$H be a Hilbert space. The recently introduced notions of the DMP inverse are extended from matrices to operators. The group, Moore---Penrose, Drazin inverses are integrated by DMP inverse and many closely equivalent relations among these inverses are investigated by using appropriate idempotents. Some new properties of DMP inverse are obtained and some known results are generalized.

Additive Results for the Generalized Drazin Inverse in a Banach Algebra

Under new conditions on Banach algebra elements a and b, we derive explicit expressions for the generalized Drazin inverse of the sum \(a+b\). As an application of our results, we present new representations for the generalized Drazin inverse of a block matrix in a Banach algebra.

Determinantal representations of the W-weighted Drazin inverse over the quaternion skew field

Within the framework of the theory of the column and row determinants, we obtain new determinantal representations of the W-weighted Drazin inverse over the quaternion skew field. We give determinantal representations of the W-weighted Drazin inverse by using previously introduced determinantal representations of the Drazin inverse, the Moore–Penrose inverse, and the limit representations of the W-weighted Drazin inverse in some special case.

Reverse order laws for the generalized Drazin inverse in Banach algebras

Under some conditions, the reverse order laws for the generalized Drazin inverse are investigated in Banach algebras. Similar results related to the Drazin inverse in a ring are presented too.

New Expressions for the Weighted Drazin Inverse of a Modified Matrix

New Expressions for the Weighted Drazin Inverse of a Modified Matrix

Infinitely many Zhang functions resulting in various ZNN models for time-varying matrix inversion with link to Drazin inverse

In this Letter, by generalizing the notion of Zhang functions (ZFs) from previous work, a novel general-form Zhang function (NGFZF) is proposed, developed and investigated. Specifically, based on the NGFZF, infinitely many ZFs (as error functions) can be readily generated by successively selecting the different values of its parameters. Besides, by employing the NGFZF, a novel general-form Zhang neural net (NGFZNN) is proposed and studied for real-time solution of a time-varying matrix inverse (also termed, Zhang matrix inverse, ZMI). Moreover, a link between ZMI and Drazin inverse is discovered and further generalized to solve for the time-varying Drazin inverse (TVDI).

Gröbner basis computation of Drazin inverses with multivariate rational function entries

In this paper we show how to apply Gröbner bases to compute the Drazin inverse of a matrix with multivariate rational functions as entries. When the coefficients of the rational functions depend on parameters, we give sufficient conditions for the Drazin inverse to specialize properly. In addition, we extend the method to weighted Drazin inverses. We present an empirical analysis that shows a good timing performance of the method.

Recurrent Neural Network for Computing the Drazin Inverse

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.

New results on common properties of the products AC and BA

Let X, Y be Banach spaces, A:X⟶Y and B,C:Y⟶X be bounded linear operators satisfying operator equation ABA=ACA. In this paper, we show that the products AC and BA share the spectral properties such as Drazin invertibility, polaroidness and B-Fredholmness. As an application, we show that generalized Weyl's theorem holds for the Aluthge transform T˜ of an algebraically (n,k)-quasiparanormal operator T. Also, Cline's formula for Drazin inverse in a ring with identity is established in the case when aba=aca, and in this case we establish Cline's formula for generalized Drazin inverse in the setting of Banach algebra.

Weighted binary relations involving the Drazin inverse

The Drazin inverse of a matrix has been used in the literature to define a pre-order on the set of square complex matrices. In this paper we analyze new binary relations defined on the set of rectangular complex matrices and some relationships to the W-support idempotent. We introduce the class of weighted Drazin equal projectors and analyze the pre-orders on this class. Moreover, adjacent matrices are studied under the considered relations. Finally, some observations on weighted partial orders are given.

Formulae for the Generalized Drazin Inverse of a Block Matrix in Banach Algebras

We present some new representations for the generalized Drazin inverse of a block matrix in a Banach algebra under conditions weaker than those used in resent papers on the subject.

Representations for the Drazin inverse of a modified matrix

In this paper expressions for the Drazin inverse of a modified matrix A - CDdB are presented in terms of the Drazin inverses of A and the generalized Schur complement D - BAdC under weaker restrictions. Our results generalize and unify several results in the literature and the Sherman-Morrison- Woodbury formula.

An Efficient Algorithm for Computing the Generalized Null Space Decomposition

The generalized null space decomposition (GNSD) is a unitary reduction of a general matrix $A$ of order $n$ to a block upper triangular form that reveals the structure of the Jordan blocks of $A$ corresponding to a zero eigenvalue. The reduction was introduced by Kublanovskaya. It was extended first by Ruhe and then by Golub and Wilkinson, who based the reduction on the singular value decomposition. Unfortunately, if $A$ has large Jordan blocks, the complexity of these algorithms can approach the order of $n^4$. This paper presents an alternative algorithm, based on repeated updates of a QR decomposition of $A$, that is guaranteed to be of order $n^{3}$. Numerical experiments confirm the stability of this algorithm, which turns out to produce essentially the same form as that of Golub and Wilkinson. The effect of errors in $A$ on the ability to recover the original structure is investigated empirically. Several applications are discussed, including the computation of the Drazin inverse.

Representations for the generalized Drazin inverse of the sum in a Banach algebra and its application for some operator matrices.

We investigate additive properties of the generalized Drazin inverse in a Banach algebra A. We find explicit expressions for the generalized Drazin inverse of the sum a + b, under new conditions on a, b ∈ A. As an application we give some new representations for the generalized Drazin inverse of an operator matrix.

On block triangular matrices with signed Drazin inverse

The sign pattern of a real matrix A, denoted by sgnA, is the (+, −, 0)-matrix obtained from A by replacing each entry by its sign. Let Q(A) denote the set of all real matrices B such that sgnB = sgnA. For a square real matrix A, the Drazin inverse of A is the unique real matrix X such that A k+1 X = A k, XAX = X and AX = XA, where k is the Drazin index of A. We say that A has signed Drazin inverse if \(\operatorname{sgn} {\tilde A^d} = \operatorname{sgn} {A^d}\) for any \(\tilde A \in Q(A)\), where A d denotes the Drazin inverse of A. In this paper, we give necessary conditions for some block triangular matrices to have signed Drazin inverse.

On additive properties of the Drazin inverse of block matrices and representations

In this paper, we give a new additive formula for the Drazin inverse under conditions weaker than those used in some current literature on this subject. Also, we obtain representations for the Drazin inverse of a complex block matrix having generalized Schur complement equal to zero.

Further Results on the Generalized Drazin Inverse of Block Matrices in Banach Algebras

The objective of this paper is to derive formulae for the generalized Drazin inverse of a block matrix in a Banach algebra \(\mathcal{A}\) under different conditions. Let \(x=\left[ \begin{array}{c@{\quad }c} a&{}b\\ c&{}d\end{array}\right] \in \mathcal{A}\) relative to the idempotent \(p\in \mathcal{A}\) and \(a\in p\mathcal{A}p\) be generalized Drazin invertible. The formulae for the generalized Drazin inverse are obtained under the more general case that the generalized Schur complement \(s=d-ca^db\) is generalized Drazin invertible, which covers the cases that \(s\) is Drazin invertible, \(s\) is group invertible, or \(s\) is equal to zero. Thus, recent results on the Drazin inverse of block matrices and block-operator matrices are extended to a more general setting.

Drazin invertibility of sum and product of closed linear operators

The paper present a survey of results concerning the fundamental properties of the Drazin inverse for bounded operators and an interesting study of the Drazin inverse for a closed operator in a Banach space. Some necessary and sufficient conditions for $A$ closed linear operator to possess a Drazin inverse $A^D$ are given, we obtain also a useful caracterization and explicit formula for the Drazin inverse $(A+B)^D$ and $(A B)^D$ if $A$ and $B$ are closed operators.

Left and right generalized Drazin invertible operators

In this paper, we define and study the left and the right generalized Drazin inverse of bounded operators in a Banach space. We show that the left (resp. the right) generalized Drazin inverse is a sum of a left invertible (resp. a right invertible) operator and a quasi-nilpotent one. In particular, we define the left and the right generalized Drazin spectra of a bounded operator and also show that these sets are compact in the complex plane and invariant under additive commuting quasi-nilpotent perturbations. Furthermore, we prove that a bounded operator is left generalized Drazin invertible if and only if its adjoint is right generalized Drazin invertible. An equivalent definition of the pseudo-Fredholm operators in terms of the left generalized Drazin invertible operators is also given. Our obtained results are used to investigate some relationships between the left and right generalized Drazin spectra and other spectra founded in Fredholm theory.