Drazin Inverse Research Articles

ADDITIVE PROPERTIES OF THE DRAZIN INVERSE FOR MATRICES AND BLOCK REPRESENTATIONS: A SURVEY

In this paper, a review of a development of the Drazin inverse for the sum of two matrices has been given. Since this topic is closely related to the problem of finding the Drazin inverse of a 2x2 block matrix, the paper also offers a survey of this subject.

Jacobson's lemma for the generalized \(n\)-strong Drazin inverses in rings and in operator algebras

In this paper, we extend Jacobson's lemma for Drazin inverses to the generalized \(n\)-strong Drazin inverses in a ring, and prove that \(1-ac\) is generalized \(n\)-strong Drazin invertible if and only if \(1-ba\) is generalized \(n\)-strong Drazin invertible, provided that \(a(ba)^{2}=abaca=acaba=(ac)^{2}a\). In addition, Jacobson's lemma for the left and right Fredholm operators, and furthermore, for consistent in invertibility spectral property and consistent in Fredholm and index spectral property are investigated.

Consensus of Singular Linear Multiagent Systems Via Hybrid Control

In this article, the consensus problem of singular linear multiagent systems (SLMASs) is investigated. Based on the properties of Drazin inverse, the considered SLMAS is transformed into an equivalent differential-algebraic system. To address this problem, an event-triggered mechanism is adopted in the existing works. However, it is pointed out in this article that this mechanism is not so effective for this problem. The reason is that the discontinuity of event-triggered controller can cause the algebraic subsystem to malfunction at the triggering instants. To avoid this phenomenon, a hybrid control strategy is adopted. The main characteristic of this strategy is that it is organized by an event-triggered controller and an impulsive controller. To show the validity of theoretical results, an example is given.

Explicit Solutions of the Singular Yang–Baxter-like Matrix Equation and Their Numerical Computation

We derive several explicit formulae for finding infinitely many solutions of the equation \(AXA=XAX\), when A is singular. We start by splitting the equation into a couple of linear matrix equations and then show how the projectors commuting with A can be used to get families containing an infinite number of solutions. Some techniques for determining those projectors are proposed, which use, in particular, the properties of the Drazin inverse, spectral projectors, the matrix sign function, and eigenvalues. We also investigate in detail how well-known similarity transformations like Jordan and Schur decompositions can be used to obtain new representations of the solutions. The computation of solutions by the suggested methods using finite precision arithmetic is also a concern. Difficulties arising in their implementation are identified and ideas to overcome them are discussed. Numerical experiments shed some light on the methods that may be promising for solving numerically the matrix equation.

Relationship between Cauchy dual and Drazin inverse of conditional type operators

In this paper, we study some parallelisms between the notion of Cauchy dual for Moore−Penrose inverse and Drazin inverse of conditional type operators on L2(Σ). In addition, we discuss measure theoretic characterizations of some classical properties for Cauchy dual and Drazin inverse of Lambert conditional operators. Moreover, some correlations between these concepts are shown.

A note on formulae for the generalized Drazin inverse of anti-triangular block operator matrices in Banach spaces

A note on formulae for the generalized Drazin inverse of anti-triangular block operator matrices in Banach spaces

New additive results for the Drazin inverse of multivalued operators

The concept of the Drazin inverse of multivalued operators in a Banach space studied by A. Ghorbel and M. Mnif [Monatsh Math. 2019;189:273–293] is generalized in the context of the generalized Drazin inverse of multivalued operators [Rocky Mountain J Math. 2020;50(4):1387–1408]. The purpose of this paper is to present new additive results for this concept. In particular, we give a sufficient condition for an everywhere defined linear relation to have at most one Drazin inverse. Some properties and the explicit expressions for the Drazin inverse of the product are obtained. Also, some results of C. Deng, H. Du [Proc Amer Math Soc. 2006;134:3309–3317] concerning the reduced minimum modulus of Drazin inverses of linear operators on Hilbert spaces are extended to the case of linear relations.

New formulae of the Drazin inverse of anti-triangular complex block matrices

Let E, F ? Cnxn. If EFiE = 0 for all i ? N, we give the explicit representation of the Drazin inverse of the block complex matrix (E I F 0). We thereby solve a wider kind of singular differential equations posed by Campbell [S.L. Campbell, The Drazin inverse and systems of second order linear differential equations, Linear & Multilinear Algebra, 14(1983), 195-198].

Well-posedness for the Chern-Simons-Schrödinger equations

<abstract><p>First, we prove uniform-in-$ \epsilon $ regularity estimates of local strong solutions to the Chern-Simons-Schrödinger equations in $ \mathbb{R}^2 $. Here $ \epsilon $ is the dispersion coefficient. Then we prove the global well-posedness of strong solutions to the limit problem $ (\epsilon = 0) $.</p></abstract>

Generalized inverses of bounded finite potent operators on Hilbert spaces

The aim of this work is to prove the existence and uniqueness of the Drazin inverse and the DMP inverses of a bounded finite potent endomorphism. In particular, we give the main properties of these generalized inverses, we offer their relationships with the adjoint operator, we study their spectrum, we compute the respective traces and determinants and we relate the Drazin inverse of a bounded finite potent operator with classical definitions of this generalized inverse. Moreover, different properties of the Moore-Penrose inverse of a bounded operator are studied.

The Sherman-Morrison-Woodbury formula for the generalized inverses

In this paper, we investigate the Sherman-Morrison-Woodbury formula for the {1}-inverses and the {2}-inverses of bounded linear operators on a Hilbert space. Some conditions are established to guarantee that (A+YGZ*)? = A? ?A?Y(G? +Z*A?Y)?Z*A? holds, where A? stands for any kind of standard inverse, {1}-inverse, {2}-inverse, Moore-Penrose inverse, Drazin inverse, group inverse, core inverse and dual core inverse of A.

On a recent result concerning the Drazin inverse of matrices

In this paper, we investigate the recent paper of Shakoor et al. [A. Shakoor, I. Ali, S. Wali, A. Rehman, Some formulas on the Drazin inverse for the sum of two matrices and block matrices, Bull. Iran. Math. Soc. 48 (2022) 351-366]. Here we prove that the main, additive result from the mentioned paper is actually a corollary of one known result. Furthermore, we give new representations for the Drazin inverse of anti-triangular block matrix, which generalize some representations from current literature on the topic.

The generalized Flanders’ theorem in unit-regular rings

Let R be a unit-regular ring, and let a, b, c ? R satisfy aba = aca. If ac or ba is Drazin invertible, we prove that their Drazin inverses are similar. Furthermore, if ac and ba are group invertible, then ac is similar to ba. For any n?n complex matrices A, B,C with ABA = ACA, we prove that AC and BA are similar if and only if their k-powers have the same rank. These generalize the known Flanders? theorem proved by Hartwig.

Generalized Drazin invertibility of the sum of two elements in a Banach algebra

In this paper, we study additive properties of the generalized Drazin inverse in a Banach algebra. We first show that a + b ? Ad under the condition that a, b ? Ad, aba? = ?a?bab?a?, and then give some explicit expressions for the generalized Drazin inverse of the sum a+b under some weaker conditions than those used in the previous papers. Some known results are extended.

Necessary optimality conditions for two-step descriptor systems

<abstract><p>This paper is concerned with an optimal control problem for general two-step descriptor systems. Based on the nonsmooth analysis and variational techniques, we establish the first-order necessary optimality conditions. Then, due to the expression of the general solution for linear descriptor systems, which is established by virtue of the Drazin inverse, we derived the generalized second-order necessary conditions for the first time. Also, for unfixed switching point case, we give some discussions.</p></abstract>

Explicit formulae for the Drazin inverse of anti-triangular block matrices

In this paper, we analyze the index relation of anti-triangular block matrices and their entries to separately obtain new and strict expressions for the Drazin inverse of them under certain circumstances. As applications, we utilize the relationship between the anti-triangular block matrix and a 2 ? 2 block matrix to establish several formulae. Our results generalize and unify a series of results in the literature.

On the Drazin inverse of anti-triangular block matrices

<abstract><p>Our aim is to present new expressions for the Drazin inverse of anti-triangular block matrices under some circumstances. Applying the established new formulae for anti-triangular block matrices, we derive explicit representations for the Drazin inverse of a $ 2\times2 $ complex block matrix under corresponding assumptions. We extend several well known results in the literature in this way.</p></abstract>

The Characterizations of WG Matrix and Its Generalized Cayley–Hamilton Theorem

Based on the core-EP decomposition, we use the WG inverse, Drazin inverse, and other inverses to give some new characterizations of the WG matrix. Furthermore, we generalize the Cayley–Hamilton theorem for special matrices including the WG matrix. Finally, we give examples to verify these results.

Drazin inverse of Lambert conditional type operators

Drazin inverse of Lambert conditional type operators

Linear maps preserving Drazin inverses of matrices over local rings

Linear maps preserving Drazin inverses of matrices over local rings