A note on the perturbation bound of the Drazin inverse

A perturbation bound for the Drazin inverse A^D with Ind(A+E)=1 has recently been developed. However, those upper bounds are not satisfied since it is not tight enough. In this paper, a sharper upper bounds for ||(A+E)^#−A^D|| with weaker conditions is derived. That new bound is also a generalization of a new general upper bound of the group inverse. We also derive a new expression of the Drazin inverse (A+E)^D with Ind(A+E)>1 and the corresponding upper bound of ||(A+E)^D−A^D|| in a special case. Numerical examples are given to illustrate the sharpness of the new bounds.     

The representation and perturbation of theW-weighted Drazin inverse

LetA andE bem x n matrices andW an n xm matrix, and letA _d,W denote the W-weighted Drazin inverse ofA. In this paper, a new representation of the W-weighted Drazin inverse ofA is given. Some new properties for the W-weighted Drazin inverseA _d,W and B_d,W are investigated, whereB =A+E. In addition, the Banach-type perturbation theorem for the W-weighted Drazin inverse ofA andB are established, and the perturbation bounds for ∥B_d,W∥ and ∥B_{d, W}, -A_d,W∥/∥A_d,W∥ are also presented. WhenA andB are square matrices andW is identity matrix, some known results in the literature related to the Drazin inverse and the group inverse are directly reduced by the results in this paper as special cases.     

New perturbation bounds for the W-weighted Drazin inverse

The constructive perturbation bounds for the W-weighted Drazin inverse are derived by two approaches in this paper. One uses the matrixG = [(A+E)W]^l?(AW)^l, whereA, E ∈ C ^mxn ,W ∈ C ^nxm ,l = max Ind(AW), Ind[(A + E)W]. The other uses a technique proposed by G. Stewart and based on perturbation theory for invariant subspaces of a matrix. The new approaches to develop perturbation bounds for W-weighted Drazin inverse of a matrix extend the previous results in [19, 29, 31, 36, 42, 44]. Several examples which indicate the sharpness of the new perturbation bounds are presented.     

Acute perturbation of Drazin inverse and oblique projectors

For an n×n complex matrix A with ind(A) = r; let A^D and A^π = I–AA^D be respectively the Drazin inverse and the eigenprojection corresponding to the eigenvalue 0 of A: For an n×n complex singular matrix B with ind(B) = s, it is said to be a stable perturbation of A, if I–(B^π–A^π)² is nonsingular, equivalently, if the matrix B satisfies the condition \(\mathcal{R}(B^s)\cap\mathcal{N}(A^r)=\left\{0\right\}\) and \(\mathcal{N}(B^s)\cap\mathcal{R}(A^r)=\left\{0\right\}\), introduced by Castro-González, Robles, and Vélez-Cerrada. In this paper, we call B an acute perturbation of A with respect to the Drazin inverse if the spectral radius ρ(B^π–A^π) < 1: We present a perturbation analysis and give suffcient and necessary conditions for a perturbation of a square matrix being acute with respect to the matrix Drazin inverse. Also, we generalize our perturbation analysis to oblique projectors. In our analysis, the spectral radius, instead of the usual spectral norm, is used. Our results include the previous results on the Drazin inverse and the group inverse as special cases and are consistent with the previous work on the spectral projections and the Moore-Penrose inverse.     

A note on the perturbation bounds of W-weighted Drazin inverse

We investigate the perturbation bound for the W-weighted Drazin inverse of a rectangular matrix and present two explicit expressions for the W-weighted Drazin inverse under the one-sided condition, which extends the results in Appl. Math. Comput. 2004;149:423–430.     

Error bounds for the perturbation of the Drazin inverse under some geometrical conditions

This paper studies the Drazin inverse for perturbed matrices. For that, given a square matrix A, we consider and characterize the class of matrices B with index s such that , and , where and denote the null space and the range space of a matrix A, respectively, and A^D denote the Drazin inverse of A. Then, we provide explicit representations for B^D and BB^D, and upper bounds for the relative error B^D-A^D/A^D and the error BB^D-AA^D. A numerical example illustrates that the obtained bounds are better than others given in the literature.     

Continuity of The Drazin inverse

Let A be an n×n matrix. It is shown that if a matrix  comes close to satisfying the definition of the Drazin inverse of A,A^D , then  is close to A^D .     

The Drazin inverse as a gradient

The coefficients in the expansion of adj(λI ? A) are expressed as gradients, and some new representations are given for the Drazin inverse of a matrix over an arbitrary field. These results are then combined to express the Drazin inverse as a gradient of a function of the entries of the matrix.     

The Drazin inverse in an arbitrary semiring

In this article, we investigate the Drazin invertibility for the elements of an arbitrary semiring. We give necessary and sufficient conditions for the existence and expressions of the Drazin inverse of an element in an arbitrary semiring. Moreover, we consider the product paq under some additional necessary conditions for which the Drazin inverse of the product paq exists.     

The Drazin inverse in multibody system dynamics

Summary The numerical analysis of multibody system dynamics is based on the equations of motion as differential-algebraic systems. A thorough analysis of the linearized equations and their solution theory leads to an equivalent system of ordinary differential equations which gives deeper insight into the derivation of integration schemes and into the stabilization approaches. The main tool is the Drazin inverse, a generalized matrix inverse, which preserves the eigenvalues. The results are illustrated by a realistic truck model. Finally, the approach is extended to the nonlinear index 2 formulation.     

The (2,2,0) Drazin inverse problem

We consider the additive Drazin problem and we study the existence of the Drazin inverse of a two by two matrix with zero (2,2) entry.     

A global error bound for quadratic perturbation of linear programs

We prove a global error bound result on the quadratic perturbation of linear programs. The error bound is stated in terms of function values.     

A note on computing the Drazin inverse

A method is given for computing the Drazin inverse of a square matrix A of order n as a polynomial in A of degree n?1 or less from the characteristic polynomial of A.     

On Drazin inverse of operator matrices

In this short paper, we offer (another) formula for the Drazin inverse of an operator matrix for which certain products of the entries vanish. We also give formula for the Drazin inverse of the sum of two operators under special conditions.     

The representations of the Drazin inverse of differences of two matrices

In this paper, we give explicit representations of (P ± Q)^D, (P ± PQ)^D and (PQ)^# of two matrices P and Q, as a function of P, Q, P^D and Q^D, under the conditions P³Q = QP and Q³P = PQ.     

A Drazin inverse for rectangular matrices

The definition of the Drazin inverse of a square matrix with complex elements is extended to rectangular matrices by showing that for any B and W,m by n and n by m, respectively, there exists a unique matrix, X, such that (BW)^k=(BW)^k+1XW for some positive integer k, XWBWX = X, and BWX = XWB. Various expressions satisfied by B, W,X and related matrices are developed.     

Representations and properties of the W-Weighted Drazin inverse

Several new representations of the W-weighted Drazin inverse are introduced. These representations are expressed in terms of various matrix powers as well as in terms of matrix products involving the Moore–Penrose inverse and the usual matrix inverse. Also, the properties of various generalized inverses which arise from derived representations are investigated. The computational complexity and efficiency of the proposed representations are considered. Representations are tested and compared among themselves in a substantial number of randomly generated test examples.