Convexity of the inverse and Moore-Penrose inverse

Convexity properties of the inverse of positive definite matrices and the Moore-Penrose inverse of nonnegative definite matrices with respect to the partial ordering induced by nonnegative definiteness are studied. For the positive definite case null-space characterizations are derived, and lead naturally to a concept of strong convexity of a matrix function, extending the conventional concept of strict convexity. The positive definite results are shown to allow for a unified analysis of problems in reproducing kernel Hilbert space theory and inequalities involving matrix means. The main results comprise a detailed study of the convexity properties of the Moore-Penrose inverse, providing extensions and generalizations of all the earlier work in this area.     

On the covariance of the Moore-Penrose inverse

Given a square complex matrix A with Moore-Penrose inverse A², we describe the class of invertible matrices T such that (TAT^-1)²=TA²T^-1.     

Line digraphs and the Moore-Penrose inverse

Various characterizations of line digraphs and of Boolean matrices possessing a Moore-Penrose inverse are used to show that a square Boolean matrix has a Moore-Penrose inverse if and only if it is the adjacency matrix of a line digraph. A similar relationship between a nonsquare Boolean matrix and a bipartite graph is also given.     

The generalized weighted Moore-Penrose inverse

In this paper, we definite a generalized weighted Moore-Penrose inverse A _M,N ⁺ of a given matrixA, and give the necessary and sufficient conditions for its existence. We also prove its uniqueness and give a representation of it. In the end we point out this generalized inverse is also a prescribed rangT and null spaceS of {2}-(or outer) inverse ofA.     

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.     

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.     

The Moore-Penrose inverse of a retrocirculant

In a recent paper Chao [2] has determined the eigenvalues of a matrix of the form A=PC where P is a permutation matrix which commutes with a certain unitary matrix and C is a circulant. Here we determine the Moore-Penrose inverse of such a “retrocirculant” and show that the nonzero eigenvalues of the Moore-Penrose inverse are the reciprocals of the nonzero eigenvalues of the retrocirculant.     

A determinantal inequality involving the Moore-Penrose inverse

Let A be a rectangular matrix of complex numbers whose rows are partitioned into r arbitrary blocks:

The Moore-Penrose inverses of each of these blocks are used to form the matrix B = (A₁⁺,…, A_r⁺). It is shown that 0 ? det (AB) ? 1. This is a generalized version of Hadamard's inequality.     

Note on an extension of the Moore-Penrose inverse

The defining equations for the Moore-Penrose inverse of a matrix are extended to give a unique type of generalized inverse for matrices over arbitrary fields.     

A numerical procedure for computing the Moore-Penrose inverse

An algorithm for computing the Moore-Penrose inverse of an arbitraryn×m real matrixA is presented which uses a Gram-Schmidt like procedure to form anA-orthogonal set of vectors which span the subspace perpendicular to the kernel ofA. This one procedure will work for any value ofn andm, and for any value of rank (A).     

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.     

Additive and multiplicative perturbation bounds for the Moore-Penrose inverse

In this paper, we obtain the additive and multiplicative perturbation bounds for the Moore-Penrose inverse under the unitarily invariant norm and the Q - norm, which improve the corresponding ones in [P.Å. Wedin, Perturbation theory for pseudo-inverses, BIT 13(1973)217-232].     

Method of elementary transformation to compute Moore-Penrose inverse

Method of elementary transformation to compute Moore-Penrose inverse is given by applying the rank equalities of matrix. The inheritable properties of Moore-Penrose inverse on rank are also discussed.     

The Moore-Penrose inverse of a morphism with factorization

Given an m-by-n matrix A of rank r over a field with an involutory automorphism, it is well known that A has a Moore-Penrose inverse if and only if rank A¹A=r= rank AA¹. By use of the full-rank factorization theorem, this result may be restated in the category of finite matrices as follows: if (A₁, r, A₂) is an (epic, monic) factorization of A:m→n through r, then A has a Moore-Penrose inverse if and only if (A¹A₁, r, A₂) and (A₁, r, A₂A¹) are, respectively, (epic, monic) factorizations of A¹A:n→n and AA¹:m→m through r. This characterization of the existence of Moore-Penrose inverses is extended to arbitrary morphisms with (epic, monic) factorizations.     

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.     

EP matrices: computation of the Moore-Penrose inverse via factorizations

EP matrices are a wide class of matrices which, among other things, can be characterized through factorizations. In this paper we are using two factorization algorithms in order to compute and factorize the Moore-Penrose inverse of a singular EP matrix. For the implementation of the algorithms we make use of a Computer Algebra System such as Maple. The results given by the proposed algorithms are faster and more accurate than the built-in Maple function in both symbolic and numerical tests, using matrices of different dimensions.     

Moore-Penrose inverse in an indefinite inner product space

The concept of the Moore-Penrose inverse in an indefinite inner product space is introduced. Extensions of some of the formulae in the Euclidean space to an indefinite inner product space are studied. In particular range-hermitianness in completely characterized. Equivalence of a weighted generalized inverse and the Moore-Penrose inverse is proved. Finally, methods of computing the Moore-Penrose inverse are presented.     

An improved method for the computation of the Moore-Penrose inverse matrix

In this article we provide a fast computational method in order to calculate the Moore-Penrose inverse of singular square matrices and of rectangular matrices. The proposed method proves to be much faster and has significantly better accuracy than the already proposed methods, while works for full and sparse matrices.