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].     

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.     

Iterative method for computing the Moore-Penrose inverse based on Penrose equations

An iterative algorithm for estimating the Moore-Penrose generalized inverse is developed. The main motive for the construction of the algorithm is simultaneous usage of Penrose equations (2) and (4). Convergence properties of the introduced method as well as their first-order and second-order error terms are considered. Numerical experiment is also presented.     

The optimal perturbation bounds of the Moore-Penrose inverse under the Frobenius norm

We obtain the optimal perturbation bounds of the Moore-Penrose inverse under the Frobenius norm by using Singular Value Decomposition, which improved the results in the earlier paper [P.-Å. Wedin, Perturbation theory for pseudo-inverses, BIT 13 (1973) 217-232]. In addition, a perturbation bound of the Moore-Penrose inverse under the Frobenius norm in the case of the multiplicative perturbation model is also given.     

The period and base of a reducible sign pattern matrix

A square sign pattern matrix A (whose entries are ) is said to be powerful if all the powers A,A²,A³,…, are unambiguously defined. For a powerful pattern A, if A^l=A^l+p with l and p minimal, then l is called the base of A and p is called the period of Li et al. [On the period and base of a sign pattern matrix, Linear Algebra Appl. 212/213 (1994) 101-120] characterized irreducible powerful sign pattern matrices. In this paper, we characterize reducible, powerful sign pattern matrices and give some new results on the period and base of a powerful sign pattern matrix.     

The inverse positivity of perturbed tridiagonal M-matrices

A well-known property of an M-matrix M is that the inverse is element-wise non-negative, which we write as M^-1?0. In this paper, we consider element-wise perturbations of non-symmetric tridiagonal M-matrices and obtain sufficient bounds on the perturbations so that the non-negative inverse persists. These bounds improve the bounds recently given by Kennedy and Haynes [Inverse positivity of perturbed tridiagonal M-matrices, Linear Algebra Appl. 430 (2009) 2312-2323]. In particular, when perturbing the second diagonals (elements (l,l+2) and (l,l-2)) of M, these sufficient bounds are shown to be the actual maximum allowable perturbations. Numerical examples are given to demonstrate the effectiveness of our estimates.     

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).     

The Drazin inverse of a singular,unreduced tridiagonal matrix

For a tridiagonal, singular matrix A   we present a method for the computation of the polynomial p(λ)$p(\lambda )$ such that A^D=p(A)$A^D=p(A)$ holds, where A^D$A^D$ is the Drazin inverse of A. The approach is based on the recursion of characteristic polynomials of leading principal submatrices of A.     

Characterization of ray pattern matrix whose determinantal region has two components after deleting the origin

Ray nonsingular matrices are generalizations of sign nonsingular matrices. The problem of characterizing ray nonsingular matrices is still open. The study of the determinantal regions R_A of ray pattern matrices is closely related to the study of ray nonsingular matrices. It was proved that if R_A?{0} is disconnected, then it is a union of two opposite open sectors (or open rays). In this paper, we characterize those ray patterns whose determinantal regions become disconnected after deleting the origin. The characterization is based on three classes (F1), (F2) and (F3) of matrices, which can further be characterized in terms of the sets of the distinct signed transversal products of their ray patterns. Moreover, we show that in the fully indecomposable case, a matrix A is in the class (F1) (or (F2), respectively) if and only if A is ray permutation equivalent to a real SNS (or non-SNS, respectively) matrix.     

Drazin inverse of partitioned matrices in terms of Banachiewicz-Schur forms

Let be a partitioned matrix, where A and D are square matrices. Denote the Drazin inverse of A by A^D. The purpose of this paper is twofold. Firstly, we develop conditions under which the Drazin inverse of M having generalized Schur complement, S=D-CA^DB, group invertible, can be expressed in terms of a matrix in the Banachiewicz-Schur form and its powers. Secondly, we deal with partitioned matrices satisfying rank(M)=rank(A^D)+rank(S^D), and give conditions under which the group inverse of M exists and a formula for its computation.     

An algorithm for determining if the inverse of a strictly diagonally dominant Stieltjes matrix is strictly ultrametric

Summary It was recently shown that the inverse of a strictly ultrametric matrix is a strictly diagonally dominant Stieltjes matrix. On the other hand, as it is well-known that the inverse of a strictly diagonally dominant Stieltjes matrix is a real symmetric matrix with nonnegative entries, it is natural to ask, conversely, if every strictly diagonally dominant Stieltjes matrix has a strictly ultrametric inverse. Examples show, however, that the converse is not true in general, i.e., there are strictly diagonally dominant Stieltjes matrices in ^n×n (for everyn3) whose inverses are not strictly ultrametric matrices. Then, the question naturally arises if one can determine which strictly diagonally dominant Stieltjes matrices, in ^n×n (n3), have inverses which are strictly ultrametric. Here, we develop an algorithm, based on graph theory, which determines if a given strictly diagonally dominant Stieltjes matrixA has a strictly ultrametric inverse, where the algorithm is applied toA and requires no computation of inverse. Moreover, if this given strictly diagonally dominant Stieltjes matrix has a strictly ultrametric inverse, our algorithm uniquely determines this inverse as a special sum of rank-one matrices.Research supported by the National Science FoundationResearch supported by the Deutsche Forschungsgemeinschaft     

Schröder matrix as inverse of Delannoy matrix

Using Riordan arrays, we introduce a generalized Delannoy matrix by weighted Delannoy numbers. It turns out that Delannoy matrix, Pascal matrix, and Fibonacci matrix are all special cases of the generalized Delannoy matrices, meanwhile Schröder matrix and Catalan matrix also arise in involving inverses of the generalized Delannoy matrices. These connections are the focus of our paper. The half of generalized Delannoy matrix is also considered. In addition, we obtain a combinatorial interpretation for the generalized Fibonacci numbers.     

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.     

Determinant of the distance matrix of a tree with matrix weights

Let T be a tree with n vertices and let D be the distance matrix of T. According to a classical result due to Graham and Pollack, the determinant of D is a function of n, but does not depend on T. We allow the edges of T to carry weights, which are square matrices of a fixed order. The distance matrix D of T is then defined in a natural way. We obtain a formula for the determinant of D, which involves only the determinants of the sum and the product of the weight matrices.     

Shorted operators of partitioned matrices and application

In this paper, our main objective is to study the effect of appending/deleting a column/row on the shorted operators. It turns out that for matrices A and B for which the shorted operator S(A|B) exists, S(A₁|B₁) of the matrix A₁=[A:a] with respect to the matrix B₁=[B:b], when it exists, is obtained by appending a suitable column to S(A|B). Moreover, if S(A₁|B₁) exists, then S(A|B) exists and is obtained from S(A₁|B₁) by dropping its last column. In the process, we study the effect of appending/deleting a column/row on the space pre-order and the parallel sum of parallel summable matrices. Finally, we specialize to the case of and matrices and study the effect of bordering (by an additional column and a row) on the shorted operator. We conclude the paper with an application to Linear Models with singular dispersion structure.     

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.     

Additive results for the Drazin inverse of block matrices and applications

In this paper, we consider the Drazin inverse of a sum of two matrices and derive additive formulas under conditions weaker than those used in some recent papers on the subject. As a corollary we get the main results from the paper of Yang and Liu [H. Yang, X. Liu, The Drazin inverse of the sum of two matrices and its applications, J. Comput. Appl. Math. 235 (2011) 1412-1417]. As an application we give some new representations for the Drazin inverse of a block matrix.     

On the inverse mean first passage matrix problem and the inverse M-matrix problem

The inverse mean first passage time problem is given a positive matrix M∈R^n,n, then when does there exist an n-state discrete-time homogeneous ergodic Markov chain C, whose mean first passage matrix is M? The inverse M-matrix problem is given a nonnegative matrix A, then when is A an inverse of an M-matrix. The main thrust of this paper is to show that the existence of a solution to one of the problems can be characterized by the existence of a solution to the other. In so doing we extend earlier results of Tetali and Fiedler.     

Primitive zero-symmetric sign pattern matrices with the maximum base

In [B. Cheng, B. Liu, The base sets of primitive zero-symmetric sign pattern matrices, Linear Algebra Appl. 428 (2008) 715-731], Cheng and Liu studied the bases of primitive zero-symmetric sign pattern matrices. The sharp upper bound of the bases was obtained. In this paper, we characterize the sign pattern matrices with the sharp bound.     

Generalizations of M-matrices which may not have a nonnegative inverse

Generalizations of M-matrices are studied, including the new class of GM-matrices. The matrices studied are of the form sI-B with B having the Perron-Frobenius property, but not necessarily being nonnegative. Results for these classes of matrices are shown, which are analogous to those known for M-matrices. Also, various splittings of a GM-matrix are studied along with conditions for their convergence.