Bounds on the inverses of nonnegative triangular matrices with monotonicity properties

Some bounds on the entries and on the norm of the inverse of triangular matrices with nonnegative and monotone entries are found. All the results are obtained by exploiting the properties of the fundamental matrix of the recurrence relation which generates the sequence of the entries of the inverse matrix. One of the results generalizes a theorem contained in a recent article of one of the authors about Toeplitz matrices.     

On the closed representation for the inverses of Hessenberg matrices

The general representation for the elements of the inverse of any Hessenberg matrix of finite order is here extended to the reduced case with a new proof. Those entries are given with proper Hessenbergians from the original matrix. It justifies both the use of linear recurrences of unbounded order for such computations on matrices of intermediate order, and some elementary properties of the inverse. These results are applied on the resolvent matrix associated to a finite Hessenberg matrix in standard form. Two examples on the unit disk are given.     

On the magic square and inverse

In this note, we give a method for finding the inverse of a three by three magic square matrix without using the usual methods for finding the inverse of a matrix. Also we give a method for finding the inverse of a three by three magic square matrix whose entries are also matrices. By using these ideas, we can construct large matrices whose inverses can be calculated easily.     

On the inverse of certain patterned sums of matrices with Kronecker product structures

In this article, we derive explicit expressions for the entries of the inverse of a patterned matrix that is a sum of Kronecker products. This matrix keeps the Kronecker structure under matrix inversion, and it is used, for example, in statistics, in particular in the linear mixed model analysis. The obtained results present new and extended existing algorithms for the inversion of the considered patterned matrices. We also obtain a closed-form inverse in terms of block matrices.     

Positive definite completions of partial Hermitian matrices

The question of which partial Hermitian matrices (some entries specified, some free) may be completed to positive definite matrices is addressed. It is shown that if the diagonal entries are specified and principal minors, composed of specified entries, are positive, then, if the undirected graph of the specified entries is chordal, a positive definite completion necessarily exists. Furthermore, if this graph is not chordal, then examples exist without positive definite completions. In case a positive definite completion exists, there is a unique matrix, in the class of all positive definite completions, whose determinant is maximal, and this matrix is the unique one whose inverse has zeros in those positions corresponding to unspecified entries in the original partial Hermitian matrix. Additional observations regarding positive definite completions are made.     

The inverse and determinant of a 2 × 2 uniformly distributed random matrix

Formulae are derived for the density of the determinant and the elements of the inverse of a 2 × 2 matrix, with entries which are independent random variables uniformlly distributed on [0,1]. Graphs of the densities are presented, and the relevance of the results to interval matrices is discussed.     

局部环上矩阵广义逆半群的子集及其性质

设R是一个局部环,A是一个可相似对角化的n阶矩阵.利用矩阵方法研究了环R上矩阵A的广义逆半群的子集,得到了其做成正规子群的条件和其中元素可逆的条件,也得到了矩阵广义逆半群的一些性质.     

Moore–Penrose inverse of an invertible infinite matrix

In this article we show that, contrary to finite matrices (with real or complex entries) an invertible infinite matrix V could have a Moore–Penrose inverse that is not a classical inverse of V. This also answers a recent open problem on infinite matrices.     

Estimates for the inverse elements of tridiagonal matrices

In this work, new upper and lower bounds for the inverse entries of the tridiagonal matrices are presented. The bounds improve the bounds in D. Kershaw [Inequalities on the elements of the inverse of a certain tridiagonal matrix, Math. Comput. 24 (1970) 155–158], P.N. Shivakumar, C.X. Ji [Upper and lower bounds for inverse elements of finite and infinite tridiagonal matrices, Linear Algebr. Appl. 247 (1996) 297–316], R. Nabben [Two-sided bounds on the inverse of diagonally dominant tridiagonal matrices, Linear Algebr. Appl. 287 (1999) 289–305] and R. Peluso, T. Politi [Some improvements for two-sided bounds on the inverse of diagonally dominant tridiagonal matrices, Linear. Algebr. Appl. 330 (2001) 1–14].     

Isospectral flow method for nonnegative inverse eigenvalue problem with prescribed structure

The nonnegative inverse eigenvalue problem is that given a family of complex numbers λ={λ₁,…,λ_n}, find a nonnegative matrix of order n with spectrum λ. This problem is difficult and remains unsolved partially. In this paper, we focus on its generalization that the reconstructed nonnegative matrices should have some prescribed entries. It is easy to see that this new problem will come back to the common nonnegative inverse eigenvalue problem if there is no constraint of the locations of entries. A numerical isospectral flow method which is developed by hybridizing the optimization theory and steepest descent method is used to study the reconstruction. Moreover, an error estimate of the numerical iteration for ordinary differential equations on the matrix manifold is presented. After that, a numerical method for the nonnegative symmetric inverse eigenvalue problem with prescribed entries and its error estimate are considered. Finally, the approaches are verified by the numerical test results.     

Level-m scaled circulant factor matrices over the complex number field and the quaternion division algebra

The level-m scaled circulant factor matrix over the complex number field is introduced. Its diagonalization and spectral decomposition and representation are discussed. An explicit formula for the entries of the inverse of a level-m scaled circulant factor matrix is presented. Finally, an algorithm for finding the inverse of such matrices over the quaternion division algebra is given.     

Pseudospectra of isospectrally reduced matrices

An isospectral matrix reduction is a procedure that reduces the size of a matrix while maintaining its eigenvalues up to a known set. As to not violate the fundamental theorem of algebra, the reduced matrices have rational functions as entries. Because isospectral reductions can preserve the spectrum of a matrix, they are fundamentally different from say the restriction of a matrix to an invariant subspace. We show that the notion of pseudospectrum can be extended to a wide class of matrices with rational function entries and that the pseudospectrum of such matrices shrinks with isospectral reductions. Hence, the eigenvalues of a reduced matrix are more robust to entry‐wise perturbations than the eigenvalues of the original matrix. Moreover, the isospectral reductions considered here are more general than those considered elsewhere. We also introduce the notion of an inverse pseudospectrum (or pseudoresonances), which indicates how stable the poles of a rational function valued matrix are to entry‐wise perturbations. Illustrations of these concepts are given for mass‐spring networks. Copyright © 2014 John Wiley & Sons, Ltd.     

Singular, nonsingular, and bounded rank completions of ACI-matrices

An affine column independent matrix is a matrix whose entries are polynomials of degree at most 1 in a number of indeterminates where no indeterminate appears with a nonzero coefficient in two different columns. A completion is a matrix obtained by giving values to each of the indeterminates. Affine column independent matrices are more general than partial matrices where each entry is either a constant or a distinct indeterminate. We determine when the rank of all completions of an affine column independent matrix is bounded by a given number, generalizing known results for partial matrices. We also characterize the square partial matrices over a field all of whose completions are nonsingular. The maximum number of free entries in such matrices of a given order is determined as well as the partial matrices with this maximum number of free entries.     

On parametrization of totally nonpositive matrices and applications

The problem of accurate computations for totally non‐negative matrices has been studied; however, it remains open for other sign regular matrices. One major obstacle is that there is no known parametrization of these matrices. The main contribution of the present work is that we provide such parametrization of nonsingular totally nonpositive matrices. A useful application of our results is that these parameters can determine accurately the entries of the inverse of a nonsingular totally nonpositive matrix. Copyright © 2011 John Wiley & Sons, Ltd.     

环Z/p~kZ上矩阵广义逆的拓展

设R=Z/pkZ(其中k>1,p是一个奇素数),A是R上一个给定的可相似对角化的n阶矩阵.利用组合方法和有限局部环上的矩阵方法,讨论了矩阵A的拓展广义逆,得到了矩阵A的拓展广义逆存在的充要条件和一些的计数定理.     

The conditioning of Toeplitz band matrices

In this paper, practical conditions to check the well-conditioning of a family of nonsingular Toeplitz band matrices are obtained. All the results are based on the location of the zeros of a polynomial associated with the given family of Toeplitz matrices.The same analysis is also used to derive uniform componentwise bounds for the entries of the inverse matrices in such family.     

A probing method for computing the diagonal of a matrix inverse

The computation of some entries of a matrix inverse arises in several important applications in practice. This paper presents a probing method for determining the diagonal of the inverse of a sparse matrix in the common situation when its inverse exhibits a decay property, i.e. when many of the entries of the inverse are small. A few simple properties of the inverse suggest a way to determine effective probing vectors based on standard graph theory results. An iterative method is then applied to solve the resulting sequence of linear systems, from which the diagonal of the matrix inverse is extracted. The results of numerical experiments are provided to demonstrate the effectiveness of the probing method. Copyright © 2011 John Wiley & Sons, Ltd.     

Characterization of symmetric M-matrices as resistive inverses

We aim here at characterizing those nonnegative matrices whose inverse is an irreducible Stieltjes matrix. Specifically, we prove that any irreducible Stieltjes matrix is a resistive inverse. To do this we consider the network defined by the off-diagonal entries of the matrix and we identify the matrix with a positive definite Schrödinger operator whose ground state is determined by the lowest eigenvalue of the matrix and the corresponding positive eigenvector. We also analyze the case in which the operator is positive semidefinite which corresponds to the study of singular irreducible symmetric M-matrices.     

Vertex degrees and doubly stochastic graph matrices

In this article, the relationship between vertex degrees and entries of the doubly stochastic graph matrix has been investigated. In particular, we present an upper bound for the main diagonal entries of a doubly stochastic graph matrix and investigate the relations between a kind of distance for graph vertices and the vertex degrees. These results are used to answer in negative Merris' question on doubly stochastic graph matrices. These results may also be used to establish relations between graph structure and entries of doubly stochastic graph matrices. © 2010 Wiley Periodicals, Inc. J Graph Theory 66:104‐114, 2011