Abstract:
The DGMRES method for solving Drazin-inverse solution of singular linear systems is generally used with restarting. But the restarting often slows down the convergence and DGMRES often stagnates. We show that adding some eigenvectors to the subspace can improve the convergence just like the method proposed by R.Morgan in [R.Morgan, A restarted GMRES method augmented with eigenvectors, SIAM J.Matrix Anal.Appl. 16 (1995)1154-1171]. We derive the implementation of this method and present some numerical examples to show the advantages of this method.

Abstract:
For the singular saddle-point problems with nonsymmetric positive definite $(1,1)$ block, we present a general constraint preconditioning (GCP) iteration method based on a singular constraint preconditioner. Using the properties of the Moore-Penrose inverse, the convergence properties of the GCP iteration method are studied. In particular, for each of the two different choices of the $(1,1)$ block of the singular constraint preconditioner, a detailed convergence condition is derived by analyzing the spectrum of the iteration matrix. Numerical experiments are used to illustrate the theoretical results and examine the effectiveness of the GCP iteration method. Moreover, the preconditioning effects of the singular constraint preconditioner for restarted generalized minimum residual (GMRES) and quasi-minimal residual (QMR) methods are also tested.

Abstract:
We investigate the generalized Drazin inverse of ？ over Banach spaces stemmed from the Drazin inverse of a modified matrix and present its expressions under some conditions.

Abstract:
By using the matrix decomposition and the reverse order law, we provide some new expressions of the Drazin inverse for any block matrix with rank constraints. 1. Introduction Let be a square complex matrix. The symbols and stand for the rank and the Moore-Penrose inverse of the matrix , respectively. The Drazin inverse of is the unique matrix satisfying where is the index of , the smallest nonnegative integer such that . We write . The Drazin inverse of a square matrix plays an important role in various fields like singular differential equations and singular difference equations, Markov chains, and iterative methods. The problem of finding explicit representations for the Drazin inverse of a complex block matrix, in terms of its blocks was posed by Campbell and Meyer [1, 2] in 1979. Many authors have considered this problem and have provided formulas for under some specific conditions [3–6]. In this paper, under rank constraints, we will present some new representations of which have not been discussed before. 2. Preliminary Lemma 2.1 (see [4]). Let and be square matrices of the same order. If , then where . If and , then . Lemma 2.2 (see [7]). Let where , are square matrices with , . Then where Lemma 2.3 (see [8]). Let , . Then if and only if , satisfy where , and with . Lemma 2.4 (see [9]). Let . Then 3. Main Results In this section, with rank equality constraints, we consider the Drazin inverse of block matrices. Let , where is invertible and is singular. It is easy to verify that can be decomposed as Let where . According to Lemma 2.3, we have the following theorem. Theorem 3.1. Let , where is invertible and is singular. If where , , , then has the following form: Proof. From Lemma 2.3 and (3.1), we know that if where , then Note that Let From Lemma 2.4, we have Note that , . Then we get Let , . Then can be rewritten as the following three matrix products: Since is nonsingular, then Thus, we have From the above equality and the condition (3.3), (3.5) is easily verified. Let , where , . It is easy to verify that the matrix can be decomposed as where is the generalized Schur complement of in . Let where . Then we have the following theorem. Theorem 3.2. If , and the matrices , satisfy then, where , . Proof. From Lemma 2.3 and (3.14), we get that if then Similar to the proof of Theorem 3.1, we derive that the rank condition (3.18) can be simplified as (3.16). Next, we will give the representation for . Let Since , , and , then . From Lemma 2.2, we get From Lemma 2.1 and the fact , it follows that Substituting in (3.19), the conclusion can be obtained.

Abstract:
Let C be an additive category. Suppose that φ and η: X→ X are two morphisms of C. If φ and η have the Drazin inverses such that φη=0, then φ+η has the Drazin inverse. If φ has the Drazin inverse φD such that 1X+φDη is invertible. We study the Drazin inverse (resp. group inverse) of f =φ+η and give the necessary and sufficient condition for fD(resp. f #}=(1X+φDη)-1φD. Finally, we extend the Huylebrouck's result from the group inverse to the Drazin inverse.

Abstract:
The Drazin inverse solutions of the matrix equations ${\rm {\bf A}}{\rm {\bf X}} = {\rm {\bf B}}$, ${\rm {\bf X}}{\rm {\bf A}} = {\rm {\bf B}}$ and ${\rm {\bf A}}{\rm {\bf X}}{\rm {\bf B}} ={\rm {\bf D}} $ are considered in this paper. We use both the determinantal representations of the Drazin inverse obtained earlier by the author and in the paper. We get analogs of the Cramer rule for the Drazin inverse solutions of these matrix equations and using their for determinantal representations of solutions of some differential matrix equations, ${\bf X}'+ {\bf A}{\bf X}={\bf B}$ and ${\bf X}'+{\bf X}{\bf A}={\bf B}$, where the matrix ${\bf A}$ is singular.

Abstract:
We deduce the explicit expressions for and of two matrices and under the conditions and . Also, we give the upper bound of . 1. Introduction The symbol stands for the set of complex matrices, and (for short ) stands for the identity matrix. For , its Drazin inverse, denoted by , is defined as the unique matrix satisfying where is the index of . In particular, if , is invertible and (see, e.g., [1–3] for details). Recall that for with , there exists an nonsingular matrix such that where is a nonsingular matrix and is nilpotent of index , and (see [1, 3]). It is well known that if is nilpotent, then . We always write . Drazin [2] proved that in associative ring when are Drazin invertible and . In [4], Hartwig et al. relaxed the condition to and put forward the expression for where . In recent years, the Drazin inverse of the sum of two matrices or operators has been extensively investigated under different conditions (see, [5–15]). For example, in [7], the conditions are and , in [9] they are and , and in [15], they are and . These results motivate us to investigate how to explicitly express the Drazin inverse of the sum under the conditions and , which are implied by the condition . The paper is organized as follows. In Section 2, we will deduce some lemmas. In Section 3, we will present the explicit expressions for and of two matrices and under the conditions and . We also give the upper bound of . 2. Some Lemmas In this section, we will make preparations for discussing the Drazin inverse of the sum of two matrices in next section. To this end, we will introduce some lemmas. The first lemma is a trivial consequence of [16, Theorem？？3.2]. Lemma 2.1. Let , and with , and define Then, Lemma 2.2. Let . If , then, for any positive integers ,(i) ,(ii) . Moreover, if , then Proof. (i)By induction, we can easily get the results.(ii)For , it is evident. Assume that, for , the equation holds, that is, . When , by (i), we have Hence, by induction, we have for any . Assume . By induction on for (2.3). Obviously, when , it holds by statement (i). Assume that it holds for , that is, . When , Hence (2.3) holds for any . Lemma 2.3. Let . Suppose that and . Then, for any positive integer , where the binomial coefficient . Moreover, if are nilpotent with and , then is nilpotent and its index is less than . Proof. We will show by induction that (2.6) holds. Trivially, (2.6) holds for . Assume that (2.6) holds for , that is, Then, for , we have, by Lemma 2.2, Hence (2.6) holds for any . If are nilpotent with and , then taking in (2.6) yields , that is, is nilpotent of

Abstract:
We present some formulae for the Drazin inverse of difference and product of idempotents in a ring. A number of results of bounded linear operators in Banach spaces are extended to the ring case.

Abstract:
In this paper we present expressions for the Drazin inverse of the generalized Schur complement $A-CD^{d}B$ in terms of the Drazin inverses of $A$ and the generalized Schur complement $D-BA^{d}C$ under less and weaker restrictions, which generalize several results in the literature and the formula of Sherman-Morrison-Woodbury type.