On the reduced signless Laplacian spectrum of a degree maximal graph

For a (simple) graph G, the signless Laplacian of G is the matrix A(G)+D(G), where A(G) is the adjacency matrix and D(G) is the diagonal matrix of vertex degrees of G; the reduced signless Laplacian of G is the matrix Δ(G)+B(G), where B(G) is the reduced adjacency matrix of G and Δ(G) is the diagonal matrix whose diagonal entries are the common degrees for vertices belonging to the same neighborhood equivalence class of G. A graph is said to be (degree) maximal if it is connected and its degree sequence is not majorized by the degree sequence of any other connected graph. For a maximal graph, we obtain a formula for the characteristic polynomial of its reduced signless Laplacian and use the formula to derive a localization result for its reduced signless Laplacian eigenvalues, and to compare the signless Laplacian spectral radii of two well-known maximal graphs. We also obtain a necessary condition for a maximal graph to have maximal signless Laplacian spectral radius among all connected graphs with given numbers of vertices and edges.     

Resonance in equations with a diagonal field

Let A be a real square matrix, and let J?$\text{R}$ be an interval not containing an eigenvalue of A. Is A–D nonsingular for all diagonal matrices D with entries d_i ∈J? This holds if A is symmetric, but is not true in general. We prove a necessary condition and indicate implications for an equation with a diagonal field.     

Equality conditions for the singular values of 3 × 3 matrices with one-point spectrum

Let Γ_a be an upper triangular 3 × 3 matrix with diagonal entries equal to a complex scalar a. Necessary and su.cient conditions are found for two of the singular values of Γ_a to be equal. These conditions are much simpler than the equality discr ? = 0, where the expression in the left-hand side is the discriminant of the characteristic polynomial ? of the matrix _Ga = Γ_aΓ_a. Understanding the behavior of singular values of Γ_a is important in the problem of finding a matrix with a triple zero eigenvalue that is closest to a given normal matrix A.     

On a theorem of Redheffer concerning diagonal stability

An important problem in system theory concerns determining whether or not a given LTI system is diagonally stable. More precisely, this problem is concerned with determining conditions on a matrix A such that there exists a diagonal matrix D with positive diagonal entries (i.e. a positive diagonal matrix), satisfying A^TD+DA=-Q<0. While this problem has attracted much attention over the past half century, two results of note stand out: (i) a result based on Theorems of the Alternative derived by Barker, Berman and Plemmons; and (ii) algebraic conditions derived by Redheffer. This paper is concerned with the second of these conditions. Our principal contribution is to show that Redheffer’s result can be obtained from the Kalman-Yacubovich-Popov lemma. We then show that this method of proof leads to natural generalisations of Redheffer’s result and we use these results to derive new conditions for diagonal and Hurwitz stability for special classes of matrices.     

G-matrices,J-orthogonal matrices,and their sign patterns

A real matrix A is a G-matrix if A is nonsingular and there exist nonsingular diagonal matrices D₁ and D₂ such that A^?T = D₁AD₂, where A^?T denotes the transpose of the inverse of A. Denote by J = diag(±1) a diagonal (signature) matrix, each of whose diagonal entries is +1 or ?1. A nonsingular real matrix Q is called J-orthogonal if Q^TJQ = J. Many connections are established between these matrices. In particular, a matrix A is a G-matrix if and only if A is diagonally (with positive diagonals) equivalent to a column permutation of a J-orthogonal matrix. An investigation into the sign patterns of the J-orthogonal matrices is initiated. It is observed that the sign patterns of the G-matrices are exactly the column permutations of the sign patterns of the J-orthogonal matrices. Some interesting constructions of certain J-orthogonal matrices are exhibited. It is shown that every symmetric staircase sign pattern matrix allows a J-orthogonal matrix. Sign potentially J-orthogonal conditions are also considered. Some examples and open questions are provided.     

Classes of general H-matrices

Let M(A) denote the comparison matrix of a square H-matrix A, that is, M(A) is an M-matrix. H-matrices such that their comparison matrices are nonsingular are well studied in the literature. In this paper, we study characterizations of H-matrices with either singular or nonsingular comparison matrices. The spectral radius of the Jacobi matrix of M(A) and the generalized diagonal dominance property are used in the characterizations. Finally, a classification of the set of general H-matrices is obtained.     

A trace inequality for M-matrices and the symmetrizability of a real matrix by a positive diagonal matrix

The question of whether a real matrix is symmetrizable via multiplication by a diagonal matrix with positive diagonal entries is reduced to the corresponding question for M-matrices and related to Hadamard products. In the process, for a nonsingular M-matrix A, it is shown that tr(A^-1A^T) ? n, with equality if and only if A is symmetric, and that the minimum eigenvalue of A^-1 ° A is ? 1 with equality in the irreducible case if and only if A is positive diagonally symmetrizable.     

Integrally Closed Domains,Minimal Polynomials,and Null Ideals of Matrices

Abstract

We show that every element of the integral closure D′ of a domain D occurs as a coefficient of the minimal polynomial of a matrix with entries in D. This answers affirmatively a question of Brewer and Richman, namely, if integrally closed domains are characterized by the property that the minimal polynomial of every square matrix with entries in D is in D[x]. It follows that a domain D is integrally closed if and only if for every matrix A with entries in D the null ideal of A, N _D (A)?=?{f?∈?D[x]?∣?f(A)?=?0} is a principal ideal of D[x].     

On conjectures involving second largest signless Laplacian eigenvalue of graphs

Let G=(V,E) be a simple graph. Denote by D(G) the diagonal matrix of its vertex degrees and by A(G) its adjacency matrix. Then the Laplacian matrix of G is L(G)=D(G)-A(G) and the signless Laplacian matrix of G is Q(G)=D(G)+A(G). In this paper we obtain a lower bound on the second largest signless Laplacian eigenvalue and an upper bound on the smallest signless Laplacian eigenvalue of G. In [5], Cvetkovi? et al. have given a series of 30 conjectures on Laplacian eigenvalues and signless Laplacian eigenvalues of G (see also [1]). Here we prove five conjectures.     

On simultaneous diagonalization of one Hermitian and one symmetric form

It is remarked that if A, B ? M_n(C), A = A^t, and B? = B^t, B positive definite, there exists a nonsingular matrix U such that (1) ū^tBU = I and (2) U^tAU is a diagonal matrix with nonnegative entries. Some related actions of the real orthogonal group and equations involving the unitary group are studied.     

Connected graphs with maximal Q-index: The one-dominating-vertex case

By the signless Laplacian of a (simple) graph G we mean the matrix Q(G)=D(G)+A(G), where A(G),D(G) denote respectively the adjacency matrix and the diagonal matrix of vertex degrees of G. For every pair of positive integers n,k, it is proved that if 3?k?n-3, then H_n,k, the graph obtained from the star K_1,n-1 by joining a vertex of degree 1 to k+1 other vertices of degree 1, is the unique connected graph that maximizes the largest signless Laplacian eigenvalue over all connected graphs with n vertices and n+k edges.     

Unicyclic graphs with bicyclic inverses

A graph is nonsingular if its adjacency matrix A(G) is nonsingular. The inverse of a nonsingular graph G is a graph whose adjacency matrix is similar to A(G)^?1 via a particular type of similarity. Let H denote the class of connected bipartite graphs with unique perfect matchings. Tifenbach and Kirkland (2009) characterized the unicyclic graphs in H which possess unicyclic inverses. We present a characterization of unicyclic graphs in H which possess bicyclic inverses.     

Regular and singular orthants of tridiagonal matrices

With respect to a tridiagonal matrix with variable diagonal vector g, an orthant is said to be regular (singular) if the matrix is nonsingular (singular) for all g in it. We give necessary and sufficient conditions for an orthant to be regular or singular. Our idea is based on observations of a simple two-by-two matrix, and all the results obtained are original and self-contained.     

Simultaneous diagonalization of rectangular matrices

A matrix D is said to be diagonal if its (i,j)th element is null whenever i and j are unequal. For a set {A_θ} of matrices A_θ of the same order, the paper gives necessary and sufficient conditions for nonsingular matrices S and T to exist, such that SA_θT = D_θ is diagonal for each matrix A_θ in the set.     

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.     

A forbidden structure of ray nonsingular matrices

A complex square matrix is called a ray nonsingular matrix (RNS matrix) if its ray pattern implies that it is nonsingular. In this paper, a necessary condition for RNS matrices is provided by showing that if A=I−A(D)$A=I-A(D)$ is ray nonsingular, then the arc weighted digraph D contains no forbidden cycle chains.     

A continuous approximation to the generalized Schur decomposition

We consider the problem of approximating the generalized Schur decomposition of a matrix pencil A − λB by a family of differentiable orthogonal transformations. It is shown that when B is nonsingular this approach is feasible and can be fully expressed as an autonomous differential system. When B is singular, we show that the location of zero diagonal entries of B affects the feasibility of such an approach, and hence we conclude that at least the QZ algorithm cannot be extended continuously.     

Congruence of Hermitian matrices by Hermitian matrices

Two Hermitian matrices A,B∈M_n(C) are said to be Hermitian-congruent if there exists a nonsingular Hermitian matrix C∈M_n(C) such that B=CAC. In this paper, we give necessary and sufficient conditions for two nonsingular simultaneously unitarily diagonalizable Hermitian matrices A and B to be Hermitian-congruent. Moreover, when A and B are Hermitian-congruent, we describe the possible inertias of the Hermitian matrices C that carry the congruence. We also give necessary and sufficient conditions for any 2-by-2 nonsingular Hermitian matrices to be Hermitian-congruent. In both of the studied cases, we show that if A and B are real and Hermitian-congruent, then they are congruent by a real symmetric matrix. Finally we note that if A and B are 2-by-2 nonsingular real symmetric matrices having the same sign pattern, then there is always a real symmetric matrix C satisfying B=CAC. Moreover, if both matrices are positive, then C can be picked with arbitrary inertia.     

Sign determinacy of M-matrix minors

We show that if A is an M-matrix for which the length of the longest simple cycle in its associated undirected graph G(A) is at most 3, then every minor of A has determined sign (nonnegative or nonpositive), independent of the magnitudes of the matrix entries. Consequently, if A and B are M-matrices such that G(A) and G(B) are subgraphs of an undirected graph with longest simple cycle at most 3, then all principal minors of AB are nonnegative.