Conditions for the positivity of determinants

Consider a matrix with positive diagonal entries, which is similar via a positive diagonal matrix to a symmetric matrix, and whose signed directed graph has the property that if a cycle and its symmetrically placed complement have the same sign, then they are both positive. We provide sufficient conditions so that A be a P-matrix, that is , a matrix whose principal minors are all positive. We further provide sufficiet conditions for an arbitrary matrix A whose (undirected) graph is subordinate to a tree, to be a P-matrix. If, in additionA is sign symmetric and its undirected graph is a tree, we obtain necessary and sufficient conditions that it be a P-matrix. We go on to consider the positive semi-definiteness of symmetric matrices whose graphs are subordinate to a given tree and discuss the convexity of the set of all such 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.     

A Hadamard product involving N-matrices

We show that for any pair M,N of n by n M-matrices, the Hadamard (entry-wise) product M°N_-1 is again an M-matrix. For a single M-matrix M, the matrix M°M^-1 is also considered.     

On Some Properties of Convex Matrix Sets Characterized by P -Matrices and Block P -Matrices

In this paper, we consider convex sets of real matrices and establish criteria characterizing these sets with respect to certain matrix properties of their elements. In particular, we deal with convex sets of P-matrices, block P-matrices and M-matrices, nonsingular and full rank matrices, as well as stable and Schur stable matrices. Our results are essentially based on the notion of a block P-matrix and extend and generalize some recently published results on this topic.     

Inverse M-matrix completion problem with zeros in the inverse completion

In this paper, we obtain an inverse M-matrix completion, with zeros in the inverse completion, of a noncombinatorially symmetric partial inverse M-matrix, when the associated graph is acyclic without specified paths or, in the other case, when the subgraph induced by the vertices of any cycle or specified path is a clique.     

The completable digraphs for the totally nonnegative completion problem

In this paper, we study the totally nonnegative completion problem when the partial totally nonnegative matrix is non-combinatorially symmetric. In general, this type of partial matrix does not have a totally nonnegative completion. Here, we give necessary and sufficient conditions for completion of a partial totally nonnegative matrix to a totally nonnegative matrix in the cases where the digraph of the off-diagonal specified entries takes certain forms as path, cycles, alternate paths, block graphs, etc., distinguishing between the monotonically and non-monotonically labeled case.     

M-matrix products having positive principal minors

Sufficient conditions are given for powers and products of M-matrices to have all principal minors positive. Several of these conditions involve directed graphs of the matrices. In particular we show that if A and B are irreducible M-matrices which have longest simple circuit of length two with A+B having no simple circuit longer than three, then the product AB has all principal minors positive.     

Stability of m-matrix products

We investigate various types of stability for powers and products of nonsingular M-matrices. Stability of the matrix powers is categorized according to the length of the longest simple circuit in the digraph of the matrix, while stability of the general products is categorized by the order of the matrices. Additional results are given regarding stability of the Hadamard product of M-matrices and for matrices whose digraph has a longest simple circuit of length two.     

Characterizations of singular irreducible M-matrices

In this note, We obtain some necessary and sufficient conditions such that a singular Z-matrix is a singular irreducible M-matrix.     

Siegmund Duality for Continuous Time Markov Chains on ℤ<Stack><Subscript>+</Subscript><Superscript>d</Superscript></Stack>

For the continuous time Markov chain with transition function P(t) on Z₊^d, we give the necessary and sufficient conditions for the existence of its Siegmund dual with transition function P(t). If Q, the q-matrix of P(t), is uniformly bounded, we show that the Siegmund dual relation can be expressed directly in terms of q-matrices, and a sufficient condition under which the Q-function is the Siegmund dual of some Q-function is also given.     

The completely positive and doubly nonnegative completion problems

We prove the following. Let G be an undirected graph. Every partially specified symmetric matrix, the graph of whose specified entries is G and each of whose fully specified submatrices is completely positive (equal to BB^T for some entrywise nonnegative matrix B), may be completed to a completely positive matrix if and only if G is a block-clique graph (a chordal graph in which distinct maximal cliques overlap in at most one vertex). The same result holds for matrices that are doubly nonnegative (entrywise nonnegative and positive semidefinite).     

Conditions for positive definiteness of M-matrices

The present paper concentrates on conditions that are necessary and sufficient for M-matrices to be positive definite. The obtained results can be used in the analysis of productivity of the Leontief input-output model.     

最大匹配的路变换图

图G的最大匹配的路变换图NM(G)是这样一个图,它以G的最大匹配为顶点,如果两个最大匹配M_1与M_2的对称差导出的图是一条路(长度没有限制),那么M_1和M_2在NM(G)中相邻.研究了这个变换图的连通性,分别得到了这个变换图是一个完全图或一棵树或一个圈的充要条件.     

Height bases, level bases, and the equality of the height and the level characteristics of an M-matrix

Let A be an M-matrix. We introduce the concepts of height basis, level basis, and height-level basis for the generalized nullspace of A. We explore the properties of such bases and of induced matrices. We use these results to prove some new conditions for the equality of the (spectral) height (Weyr) characteristic and the (graph theoretic) level characteristic of A, and to simplify proofs of known conditions. We also prove the existence of a Jordan basis for the generalized nullspace with all chains of maximal length nonnegative.     

Necessary conditions and a sufficient condition for the Fischer-Hadamard inequalities

It is well known that a matrix, all of whose principal minors are positive, satisfies the Fischer-Hadamard inequalities if and only if it is weakly sign symmetric. In this paper we consider the general case of matrices whose principal minors may be nonpositive. Necessary conditions and a sufficient condition for the Fischer-Hadamard inequalities to hold are given in the general case.     

The Totally Non-positive Matrix Completion Problem

In this paper, the totally non-positive matrix is introduced. The totally non-positive completion asks which partial totally non-positive matrices have a completion to a totally non-positive matrix. This problem has. in general, a negative answer. Therefore, our question is for what kind of labeled graphs G each partial totally non-positive matrix whose associated graph is G has a totally non-positive completion? If G is not a monotonically labeled graph or monotonically labeled cycle, we give necessary and sufficient conditions that guarantee the existence of the desired completion.     

Bi-block positive semidefiniteness of bi-block symmetric tensors

The positive definiteness of elasticity tensors plays an important role in the elasticity theory.In this paper,we consider the bi-block symmetric tensors,which contain elasticity tensors as a subclass.First,we define the bi-block M-eigenvalue of a bi-block symmetric tensor,and show that a bi-block symmetric tensor is bi-block positive(semi)definite if and only if its smallest bi-block M-eigenvalue is(nonnegative)positive.Then,we discuss the distribution of bi-block M-eigenvalues,by which we get a sufficient condition for judging bi-block positive(semi)definiteness of the bi-block symmetric tensor involved.Particularly,we show that several classes of bi-block symmetric tensors are bi-block positive definite or bi-block positive semidefinite,including bi-block(strictly)diagonally dominant symmetric tensors and bi-block symmetric(B)B0-tensors.These give easily checkable sufficient conditions for judging bi-block positive(semi)definiteness of a bi-block symmetric tensor.As a byproduct,we also obtain two easily checkable sufficient conditions for the strong ellipticity of elasticity tensors.     

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.     

Inertia possibilities for completions of partial hermitian matrices *

A partial Hermitian matrix is one in which some entries are specified and others are considered to be free (complex) variables. Assuming the undirected graph of the specified entries is chordal, it is shown that, with certain mild restrictions, a partial Hermitian matrix may be completed to a Hermitian matrix with any inertia allowed by the specified principal submatrices through the interlacing inequalities. This generalizes earlier work dealing with the existence of positive definite completions, and. as before, the chordality assumption is, in general, necessary. Further related observations dealing with Toeplitz completions and the minimum eigenvalues of completions are also made, and these raise additional questions.