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1.
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 Bd,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 ∥Bd,W∥ and ∥Bd, W, -Ad,W∥/∥Ad,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.  相似文献   

2.
Let and be a perturbed eigenpair of a diagonalisable matrixA. The problem is to bound the error in and . We present one absolute perturbation bound and two relative perturbation bounds. The absolute perturbation bound is an extension of Davis and Kahan's sin θ Theorem from Hermitian to diagonalisable matrices. The two relative perturbation bounds assume that and are an exact eigenpair of a perturbed matrixD 1 AD 2 , whereD 1 andD 2 are non-singular, butD 1 AD 2 is not necessarily diagonalisable. We derive a bound on the relative error in and a sin θ theorem based on a relative eigenvalue separation. The perturbation bounds contain both the deviation ofD 1 andD 2 from similarity and the deviation ofD 2 from identity. This work was partially supported by NSF grant CCR-9400921.  相似文献   

3.
A perturbation bound for the generalized polar decomposition   总被引:11,自引:0,他引:11  
LetA be anm×n complex matrix. A decompositionA=QH is termed ageneralized polar decomposition ofA ifQ is anm×n subunitary matrix (sometimes also called a partial isometry) andH a positive semidefinite Hermitian matrix. It was proved that a nonzero matrixA m×n has a unique generalized polar decompositionA=QH with the property (Q H )=(H), whereQ H denotes the conjugate transpose ofQ and (H) the column space ofH. The main result of this note is a perturbation bound forQ whenA is perturbed.  相似文献   

4.
Summary In this paper we investigate the properties of the Chebyshev solutions of the linear matrix equationAX+YB=C, whereA, B andC are given matrices of dimensionsm×r, s×n andm×n, respectively, wherer ands. We separately consider two particular cases. In the first case we assumem=r+1 andn=s+1, in the second caser=s=1 andm, n are arbitrary. For these two cases, under the assumption that the matricesA andB are full rank, we formulate necessary and sufficient conditions characterizing the Chebyshev solution ofAX+YB=C and we give the formulas for the Chebyshev error. Then, we propose an algorithm which may be applied to compute the Chebyshev solution ofAX+YB=C for some particular cases. Some numerical examples are also given.  相似文献   

5.
We consider some problems concerning generalizations of embeddings of acyclic digraphs inton-dimensional dicubes. In particular, we define an injectioni from a digraphD into then-dimensional dicubeQ n to be animmersion if for any two elementsa andb inD there is a directed path inD froma tob iff there is a directed path inQ n fromi(a) toi(b). We further define the immersion to bestrong iff there is a way of choosing these paths so that paths inQ n corresponding to arcs inD have disjoint interiors, and we introduce a natural notion of “minimality” on the set of arcs of a digraph in terms of its paths. Our main theorem then becomes:Every (minimal) n-element acyclic digraph can be (strongly) immersed in Q n. We also present examples ofn-element digraphs which cannot be immersed inQ n?1 and examples of 9n-element non-minimal digraphs which cannot be strongly immersed inQ10n ?1. We conclude with some applications.  相似文献   

6.
Let Mm,n(B) be the semimodule of all m×n Boolean matrices where B is the Boolean algebra with two elements. Let k be a positive integer such that 2?k?min(m,n). Let B(m,n,k) denote the subsemimodule of Mm,n(B) spanned by the set of all rank k matrices. We show that if T is a bijective linear mapping on B(m,n,k), then there exist permutation matrices P and Q such that T(A)=PAQ for all AB(m,n,k) or m=n and T(A)=PAtQ for all AB(m,n,k). This result follows from a more general theorem we prove concerning the structure of linear mappings on B(m,n,k) that preserve both the weight of each matrix and rank one matrices of weight k2. Here the weight of a Boolean matrix is the number of its nonzero entries.  相似文献   

7.
The Kantorovi? operators of second order are introduced byQ n f= =(B n+2 F)″ whereF is the double indefinite integraloff andB n+2 the (n+2)-th Bernstein operator. The operatorsQ n will reveal a close affinity to the so-called modified Bernstein operatorsC n introduced bySchnabl [10] on a quite different way. The article contains investigations concerning the asymptotic behavior ofQ n kn f (asn → ∞), where (k n) is a sequence of natural numbers.  相似文献   

8.
Let A be an m ×n real matrix with singular values σ1 ? ··· ? σn?1 ? σn ? 0. In cases where σn ? 0, the corresponding right singular vector υn is a natural choice to use for an approximate null vector ofA. Using an elementary perturbation analysis, we show that κ = σ1/(σn?1 ? σn) provides a quantitative measure of the intrinsic conditioning of the computation of υn from A.  相似文献   

9.
Perturbation bounds on the polar decomposition   总被引:7,自引:0,他引:7  
The polar decomposition of ann ×n-matrixA takes the formA=MH whereM is orthogonal andH is symmetric and positive semidefinite. This paper presents strict bounds, (with no order terms), on the perturbationsM,H ofM andH respectively, whenA is perturbed byA. The bounds onM can also be applied to the orthogonal Procrustes problem.  相似文献   

10.
In this paper we consider the following problem: Given two matricesA,Z∈? n×n , does there exist an invertiblen×n-matrixS such thatS ?1 AS is an upper triangular matrix andS ?1 ZS is a lower triangular matrix, and if so, what can be said about the order in which the eigenvalues ofA andZ appear on the diagonals of these triangular matrices? For special choices ofA andZ a complete solution is possible, as has been shown by several authors. Here we follow a lead, provided by Shmuel Friedland, who discussed the case where bothA andZ have at leastn-1 linearly independent eigenvectors, and we descibe the problem in terms of Jordan chains and left-Jordan chains for the matricesA, Z. The results give some insight in the question why certain classes of matrices (like the nonderogatory and the rank 1 matrices) allow for a detailed solution of the problems described above; for some of these classes the result of this analysis is presented here for the first time.  相似文献   

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