关于四个三幂等阵线性组合的可逆性和群逆

本文研究了四个三幂等阵线性组合的可逆性及群逆.利用矩阵分解的方法,获得了它们可逆及群逆的一些条件,并得到其逆和群逆的计算公式,这些结论完善了k幂等阵可逆性理论.     

两个幂等矩阵的组合的群逆

本文研究了当P与Q是两个复数域上的n阶幂等矩阵且满足PQP=PQ时,组合aP+bQ+cP Q+dQP+eQP Q的群逆问题,利用矩阵的分块及群逆的性质,证明了它是群逆阵,并且给出了其群逆的表达式,其中ab=0,a,b,c,d,e为复数.     

三幂等符号模式矩阵的结构

Abstract. A matrix whose entries are , -, and 0 is called a sign pattern matrix. For a signpattern matrix A,if A3 =A, then A is said to be sign tripotent. In this paper, the characteriza-tion of the n by n(n≥2) sign pattern matrices A which are sign tripotent has been given out.Furthermore, the necessary and sufficient condition of A3=A but A2≠A is obtained, too.     

The Maximal and Minimal Ranks of Some Expressions of Generalized Inverses of Matrices

In this paper, we first determine the maximal and minimal ranks of A — BXC with respect to X. Using those results, we then find the maximal and minimal ranks of the expressions AA^– — A^– A — BB^– A — AC^– C and B^– BA — ACC^– with respect to the choice of generalized inverses A^–, B^– and C^–. In particular, we consider the commutativity of A and A^–, A^k and A^–.The research of the author was supported in part by the Natural Sciences and Engineering Research Council of Canada.     

Properties of Hadamard product of inverse M‐matrices

This paper concerns with the properties of Hadamard product of inverse M‐matrices. Structures of tridiagonal inverse M‐matrices and Hessenberg inverse M‐matrices are analysed. It is proved that the product A ○ A^T satisfies Willoughby's necessary conditions for being an inverse M‐matrix when A is an irreducible inverse M‐matrix. It is also proved that when A is either a Hessenberg inverse M‐matrix or a tridiagonal inverse M‐matrix then A ○ A^T is an inverse M‐matrix. Based on these results, the conjecture that A ○ A^T is an inverse M‐matrix when A is an inverse M‐matrix is made. Unfortunately, the conjecture is not true. Copyright © 2004 John Wiley Sons, Ltd.     

Additive property of Drazin invertibility of elements in a ring

In this article, we investigate additive properties on the Drazin inverse of elements in rings. Under the commutative condition of ab?=?ba, we show that a?+?b is Drazin invertible if and only if 1?+?a ^D b is Drazin invertible. Not only the explicit representations of the Drazin inverse (a?+?b) ^D in terms of a, a ^D , b and b ^D , but also (1?+?a ^D b) ^D is given. Further, the same property is inherited by the generalized Drazin invertibility in a Banach algebra and is extended to bounded linear operators.     

On determinantal representation for the generalized inverse A and its applications

In this paper, first we establish a determinantal representation for the group inverse A_g of a square matrix A. Based on this, a determinantal representation for the generalized inverse A is presented. As an application, we give a determinantal formula for the unique solution of the general restricted linear system: Ax=b(x ∈ T, b ∈ AT and dim(AT)=dim(T)), which reduces to the common Cramer rule if A is non‐singular. These results extend our earlier work. Copyright © 2006 John Wiley & Sons, Ltd.     

The group inverse of the combinations of two idempotent matrices

In this article, we discuss the group inverse of aP + bQ + cPQ + dQP + ePQP + fQPQ + gPQPQ of idempotent matrices P and Q, where a, b, c, d, e, f, g ∈ ? and a ≠ 0, b ≠ 0, put forward its explicit expressions, and some necessary and sufficient conditions for the existence of the group inverse of aP + bQ + cPQ.     

Moore-Penrose Inverse of a Gram Matrix and Its Nonnegativity

Let M=[A  a] be a matrix of order m×n, where A∈ℝ ^m×(n−1) and a∈ℝ ^m is an m×1 vector. In this article, we derive a formula for the Moore-Penrose inverse of M ^* M and obtain sufficient conditions for its nonnegativity. The results presented here generalize the ones known earlier. The authors thank the anonymous referee for suggestions on an earlier version that have resulted in an improved presentation.     

线性空间中代数广义逆的最简表示

本文主要在一般线性空间框架中从纯代数的角度研究代数广义逆的可加性与表示问题.首先在线性空间中利用空间代数直和分解给出I+AT~+可逆的充要条件,进而T~+=T~+(I+A~T+)~(-1),给出了T~+具有最简表示的一系列充要条件.其次讨论了在Banach空间广义逆和Hilbert空间Moore-Penrose逆扰动问题研究中的应用.本文的主要结果推广和改进了相关文献中的一些近期成果.     

Cholesky decomposition of a positive semidefinite matrix with known kernel

The Cholesky decomposition of a symmetric positive semidefinite matrix A is a useful tool for solving the related consistent system of linear equations or evaluating the action of a generalized inverse, especially when A is relatively large and sparse. To use the Cholesky decomposition effectively, it is necessary to identify reliably the positions of zero rows or columns of the factors and to choose these positions so that the nonsingular submatrix of A of the maximal rank is reasonably conditioned. The point of this note is to show how to exploit information about the kernel of A to accomplish both tasks. The results are illustrated by numerical experiments.     

On constrained generalized inverses of matrices and their properties

A matrix G is called a generalized inverse (g-invserse) of matrix A if AGA = A and is denoted by G = A ⁻. Constrained g-inverses of A are defined through some matrix expressions like E(AE)⁻, (FA)⁻ F and E(FAE)⁻ F. In this paper, we derive a variety of properties of these constrained g-inverses by making use of the matrix rank method. As applications, we give some results on g-inverses of block matrices, and weighted least-squares estimators for the general linear model.     

More on extremal ranks of the matrix expressions A − BX ± X*B* with statistical applications

Through a Hermitian‐type (skew‐Hermitian‐type) singular value decomposition for pair of matrices (A, B) introduced by Zha (Linear Algebra Appl. 1996; 240 :199–205), where A is Hermitian (skew‐Hermitian), we show how to find a Hermitian (skew‐Hermitian) matrix X such that the matrix expressions A ? BX ± X^*B^* achieve their maximal and minimal possible ranks, respectively. For the consistent matrix equations BX ± X^*B^* = A, we give general solutions through the two kinds of generalized singular value decompositions. As applications to the general linear model {y, Xβ, σ²V}, we discuss the existence of a symmetric matrix G such that Gy is the weighted least‐squares estimator and the best linear unbiased estimator of Xβ, respectively. Copyright © 2007 John Wiley & Sons, Ltd.     

Distributions of eigenvalues and inertias of some block Hermitian matrices consisting of orthogonal projectors

A complex square matrix A is called an orthogonal projector if A ²?=?A?=?A*, where A* is the conjugate transpose of A. In this article, we first give some formulas for calculating the distributions of real eigenvalues of a linear combination of two orthogonal projectors. Then, we establish various expansion formulas for calculating the inertias, ranks and signatures of some 2?×?2 and 3?×?3, as well as k?×?k block Hermitian matrices consisting of two orthogonal projectors. Many applications of the formulas are presented in characterizing interval distributions of numbers of eigenvalues, and nonsingularity of these block Hermitian matrices. In addition, necessary and sufficient conditions are given for various equalities and inequalities of these block Hermitian matrices to hold.     

Nonsingularity and group invertibility of linear combinations of two k-potent matrices

An n × n complex matrix A is said to be k-potent if A ^k = A. Let T ₁ and T ₂ be k-potent and c ₁ and c ₂ be two nonzero complex numbers. We study the range space, null space, nonsingularity and group invertibility of linear combinations T = c ₁ T ₁ + c ₂ T ₂ of two k-potent matrices T ₁ and T ₂.     

First Alexandroff Decomposition Theorem for Topological Lattice Group Valued Measures

Let (X, F) be an Alexandroff space, let A(F) be the Boolean subalgebra of 2 ^X generated by F, let G be a Hausdorff commutative topological lattice group and let rba_F(A(F), G) denote the set of all order bounded F-inner regular finitely additive set functions from A(F) into G. Using some special properties of the elements of rba_F(A(F), G), we extend to this setting the first decomposition theorem of Alexandroff.     

Dynamical Systems Associated with Crossed Products

In this paper, we consider both algebraic crossed products of commutative complex algebras A with the integers under an automorphism of A, and Banach algebra crossed products of commutative C ^*-algebras A with the integers under an automorphism of A. We investigate, in particular, connections between algebraic properties of these crossed products and topological properties of naturally associated dynamical systems. For example, we draw conclusions about the ideal structure of the crossed product by investigating the dynamics of such a system. To begin with, we recall results in this direction in the context of an algebraic crossed product and give simplified proofs of generalizations of some of these results. We also investigate new questions, for example about ideal intersection properties of algebras properly between the coefficient algebra A and its commutant A′. Furthermore, we introduce a Banach algebra crossed product and study the relation between the structure of this algebra and the topological dynamics of a naturally associated system.     

The invertibility of the difference and the sum of commuting generalized and hypergeneralized projectors

In this article, we study necessary and sufficient conditions for the invertibility of a linear combination c ₁ A ^k ?+?c ₂ B ^l , in the case when A and B are both commuting generalized or hypergeneralized projectors. We present some results relating different matrix partial orderings and the invertibility of a linear combination c ₁ A ^k ?+?c ₂ B ^l when A and B are hypergeneralized projectors.     

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.