二元Boolean矩阵的加权Moore-Penrose逆

主要研究了二元Boolean矩阵A的加权Moore-Penrose逆的存在性问题,给出了二元Boolean矩阵A的加权Moore-Penrose逆存在的一些充分必要条件,并讨论了加权Moore-Penrose逆存在时的若干等价刻画及惟一性问题.     

Conjugate-gradient method for computing the Moore-Penrose inverse and rank of a matrix

A conjugate-gradient method is developed for computing the Moore-Penrose generalized inverseA of a matrix and the associated projectors, by using the least-square characteristics of both the method and the inverseA . Two dual algorithms are introduced for computing the least-square and the minimum-norm generalized inverses, as well asA . It is shown that (i) these algorithms converge for any starting approximation; (ii) if they are started from the zero matrix, they converge toA ; and (iii) the trace of a sequence of approximations multiplied byA is a monotone increasing function converging to the rank ofA. A practical way of compensating the self-correcting feature in the computation ofA is devised by using the duality of the algorithms. Comparison with Ben-Israel's method is made through numerical examples. The conjugate-gradient method has an advantage over Ben-Israel's method.After having completed the present paper, the author received from Professor M. R. Hestenes his paper entitledPseudo Inverses and Conjugate Gradients. This paper treated the same subject and appeared in Communications of the ACM, Vol. 18, pp. 40–43, 1975.     

Reverse order law for the Moore-Penrose inverse

In this paper we present new results related to the reverse order law for the Moore-Penrose inverse of operators on Hilbert spaces. Some finite-dimensional results are extended to infinite-dimensional settings.     

Moore-penrose inverse of the incidence matrix of a tree

Let T be a tree with n vertices, where each edge is given an orientation, and let Q be its vertex-edge incidence matrix. It is shown that the Moore-Penrose inverse of Q is the (n-1)× n matrix M obtained as follows. The rows and the columns of M are indexed by the edges and the vertices of T respectively. If e,ν are an edge and a vertex of T respectively, then the (e,ν)-entry of M is, upto a sign, the number of vertices in the connected component of T\e which does not contain ν. Furthermore, the sign of the entry is positive or negative, depending on whether e is oriented away from or towards ν. This result is then used to obtain an expression for the Moore-Penrose inverse of the incidence matrix of an arbitrary directed graph. A recent result due to Moon is also derived as a consequence.     

分块矩阵的Moore-Penrose逆

该文研究了两类3×3分块矩阵M1=AOOBCODEF,M2=ABCDEFGHK的Moore-Penrose逆的表达式,并给出了表达式成立时的条件.     

A Family of higher-order convergent iterative methods for computing the Moore-Penrose inverse

In this paper, we extend the iterative method for computing the inner inverse of a matrix proposed in Li and Li [W.G. Li, Z. Li, A family of iterative methods for computing the approximate inverse of a square matrix and inner inverse of a non-square matrix, Applied Mathematics and Computation 215 (2010) 3433-3442] to compute the Moore-Penrose inverse of a matrix, and show that the generated sequence converges to the Moore-Penrose inverse of a matrix in a higher order. The performance of the method is tested on some randomly generated matrices.     

Fuzzy矩阵的加权Moore-Penrose逆

给出了Fuzzy矩阵加权Moore-Penrose逆AM+N的定义,研究了Fuzzy矩阵加权Moore-Penrose逆AM+N的存在性问题,证明了当权矩阵M,N满足一定条件时,AM+N存在且A+MN=AT的充要条件是ANATMA≤A,推广了Fuzzy矩阵和Boolean矩阵的相应结果.     

Some identities for Moore-Penrose inverses of matrix products

A group of identities are established for the Moore-Penrose inverses and the weighted Moore-Penrose inverses of matrix products AB and ABC. Some consequences and applications are also presented.     

Norm estimations for perturbations of the weighted Moore-Penrose inverse

For a complex matrix $A\in \mathbb{C}^{m\times n}$, the relationship between the weighted Moore-Penrose inverse $A^\dag_{M_1N_1}$ and $A^\dag_{M_2N_2}$ is studied, and an important formula is derived,where $M_1\in \mathbb{C}^{m\times m}, N_1\in\mathbb{C}^{n\times n}$ and $M_2\in \mathbb{C}^{m\times m}, N_2\in\mathbb{C}^{n\times n}$ are different pair of positive definite hermitian matrices. Based on this formula, this paper initiates the study of the perturbation estimations for $A^\dag_{MN}$ in the case that $A$ is fixed, whereas both $M$ and $N$ are variable. The obtained norm upper bounds are then applied to the perturbation estimations for the solutions to the weighted linear least squares problems.     

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.     

关于布尔矩阵的广义Moore-Penrose逆

讨论布尔矩阵的广义Moore-Penrose逆.给出了一些广义Moore-Penrose逆存在的充要条件以及广义Moore-Penrose逆的一些刻划.     

Moore-Penrose Inverses and Group Inverses of Block k-Circulant Matrices

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具有广义分解的态射的广义Moore-Penrose逆

研究了预加范畴中具有广义分解的态射的广义Moore—Penrose逆,并给出了广义Moore—Penrose逆存在的充要条件及其表达式．     

Application of Moore-Penrose inverse in deciding the minimal martingale measure

The Moore-Penrose inverse is an important tool in algebra.This paper shows that the MoorePenrose inverse is also an effcient technique in determining the minimal martingale measure if a security price follows a semi-martingale which satisfies some structure condition.We extend a result of Dzhaparidze and Spreij concerning the Moore-Penrose inverse to the case that the Moore-Penrose inverse of any matrix-valued predictable process is still predictable.Furthermore,we obtain an explicit formula of the minimal martingale measure by employing the Moore-Penrose inverse.Specifically,the minimal martingale measure in a generalized Black-Scholes model is found.     

The Moore-Penrose inverse for sums of matrices under rank additivity conditions

Some results on the Moore-Penrose inverse for sums of matrices under rank additivity conditions are revisited and some new consequences are presented. Their extensions to the weighted Moore-Penrose inverse of sums of matrices under rank additivity conditions are also considered.     

Further results on the reverse order law for the Moore-Penrose inverse in rings with involution

We present some equivalent conditions of the reverse order law for the Moore-Penrose inverse in rings with involution, extending some well-known results to more general settings. Then we apply this result to obtain a set of equivalent conditions to the reverse order rule for the weighted Moore-Penrose inverse in C^∗-algebras.     

环上矩阵广义Moore-Penrose逆的存在性

讨论带有对合反自同构*有单位元的结合环R上矩阵的广义Moore-Penrose 逆,给出了环R上矩阵的广义Moore-Penrose逆存在的几个充要条件．特别,得到了环 R上矩阵A的关于M和N的广义Moore-Penrose逆存在的充要条件是A有分解A= GDH,其中D2=D,(MD)*=MD,(GD)*MGD+M(I-D)和DHN-1(DH)*+ (I-D)M-1均可逆．     

Approximation Theorems of Moore-Penrose Inverse by Outer Inverses

1 Introduction and preliminaries Let X and Y be two Hilbert spaces and T a bounded linear operator from X into Y . We use D(T ), N(T ) and R(T ), respectively, to denote the domain, null space and range of T . Recall that a linear operator T # : Y → X is…     

A one parameter method for the matrix inverse square root

This paper is motivated by the paper [3], where an iterative method for the computation of a matrix inverse square root was considered. We suggest a generalization of the method in [3]. We give some sufficient conditions for the convergence of this method, and its numerical stabillity property is investigated. Numerical examples showing that sometimes our generalization converges faster than the methods in [3] are presented.     

Nonnegative Moore-Penrose Inverse of Gram Matrices in an Indefinite Inner Product Space

In this paper, we derive necessary and sufficient conditions for the nonnegativity of the Moore-Penrose inverse of a Gram matrix defined in an indefinite inner product space using indefinite matrix multiplication. These conditions include the acuteness of a pair of closed convex cones.