Inverse M-matrices, II

This is an update of the 1981 survey by the first author. In the meantime, a considerable amount has been learned about the very special structure of the important class of inverse M-matrices. Developments since the earlier survey are emphasized, but we have tried to be somewhat complete; and, some results have not previously been published. Some proofs are given where appropriate and references are given for others. After some elementary preliminaries, results are grouped by certain natural categories.     

Two properties of M-matrices

If A is an M-matrix with the property that some power of A is lower triangular, then A is lower triangular. An analogue of the Minkowski determinant theorem is proved for a subclass of the M-matrices.     

M-matrix characterizations.I—nonsingular M-matrices

The purpose of this survey is to classify systematically a widely ranging list of characterizations of nonsingular M-matrices from the economics and mathematics literatures. These characterizations are grouped together in terms of their relations to the properties of (1) positivity of principal minors, (2) inverse-positivity and splittings, (3) stability and (4) semipositivity and diagonal dominance. A list of forty equivalent conditions is given for a square matrix A with nonpositive off-diagonal entries to be a nonsingular M-matrix. These conditions are grouped into classes in order to identify those that are equivalent for arbitrary real matrices A. In addition, other remarks relating nonsingular M-matrices to certain complex matrices are made, and the recent literature on these general topics is surveyed.     

Factorization of symmetric singular M-matrices

Suppose A is a symmetric, singular M-matrix. A sufficient condition for A to have a triangular, singular M-matrix factorization is given, and it is shown that PAP^T always has such a factorization for a particular permutation matrix P.     

On square roots of M-matrices

The question of the existence and uniqueness of an M-matrix which is a square root of an M-matrix is discussed. The results are then used to derive some new necessary and sufficient conditions for a real matrix with nonpositive off diagonal elements to be an M-matrix.     

Singular M-matrices and inverse positivity

For a matrix decomposable as A=sI?B, where B?0, it is well known that A^?1?0 if and only if the spectral radius ρ(B)>s. An extension of this result to the singular case ρ(B)=s is made by replacing A^?1 by [A+t(I?AA^D)]^?1, where A^D is the Drazin generalized inverse.     

M-matrices leading to semiconvergent splittings

An M-matrix as defined by Ostrowski is a matrix that can be split into A = sI ? B, s > 0, B ? 0 with s ? ρ(B), the spectral radius of B. M-matrices with the property that the powers of T = (1/s)B converge for some s are studied and are characterized here in terms of the nonnegativity of the group generalized inverse of A on the range space of A, extending the well-known property that A^{? 1} ? 0 whenever A is nonsingular.     

Completely mixed games and M-matrices

Any non-singular M-matrix is a completely mixed matrix game with positive value. We exploit this property to give game-theoretic proofs of several well-known characterizations of such matrices. The same methods yield also many theorems on S₀-irreducible matrices that are closely related to M-matrices.     

Tridiagonal and upper triangular inverse M-matrices

In this paper we characterize the nonnegative nonsingular tridiagonal matrices belonging to the class of inverse M-matrices. We give a geometric equivalence for a nonnegative nonsingular upper triangular matrix to be in this class. This equivalence is extended to include some reducible matrices.     

A note on factorizations of singular M-matrices

Supposing that M is a singular M-matrix, we show that there exists a permutation matrix P such that PMP^T = LU, where L is a lower triangular M-matrix and U is an upper triangular singular M-matrix. An example is given to illustrate that the above result is the best possible one.     

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.     

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.     

On algebraic Riccati equations associated with M-matrices

We consider the algebraic Riccati equation for which the four coefficient matrices form an M-matrix K. When K is a nonsingular M-matrix or an irreducible singular M-matrix, the Riccati equation is known to have a minimal nonnegative solution and several efficient methods are available to find this solution. In this paper we are mainly interested in the case where K is a reducible singular M-matrix. Under a regularity assumption on the M-matrix K, we show that the Riccati equation still has a minimal nonnegative solution. We also study the properties of this particular solution and explain how the solution can be found by existing methods.     

Characterizations of inverse M-matrices with special zero patterns

In this paper, we provide some characterizations of inverse M-matrices with special zero patterns. In particular, we give necessary and sufficient conditions for k-diagonal matrices and symmetric k-diagonal matrices to be inverse M-matrices. In addition, results for triadic matrices, tridiagonal matrices and symmetric 5-diagonal matrices are presented as corollaries.     

Matrices with a transitive graph and inverse M-matrices

The relationship between inverse M-matrices and matrices whose graph is transitive is studied. The results are applied to obtain a new proof of the characterization, due to M. Lewin and M. Neumann, of (0,1) inverse M-matrices.     

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.     

On classes of matrices containing M-matrices and hermitian positive semidefinite matrices

Two new classes of matrices are introduced, containing hermitian positive semi-definite matrices and M-matrices. The relation to other well-known classes such as ω and τ-matrices and weakly sign symmetric matrices is examined, and invariance properties are shown.     

The Kahan S.O.R. convergence bound for nonsingular and irreducible M-matrices

Let A be a nonsingular M-matrix, and let π be a block partitioning of A such that the diagonal blocks are square. Denote by J^A_π and

^A_πω the block Jacobi and the block S.O.R. iteration matrices arising from and associated with the partitioning π of A, respectively. In [5] Kahan showed that ρ (

^A_πω<1 for all 0<ω<ω':=2/[1+ρ(J^A_π)]. Under the assumption that J^A_π is irreducible we examine the question of when ρ(

^A_πω')=1 for certain recurring M-matrices.