Zero-nonzero Pattern Research Articles

Symmetric nonnegative trifactorization of pattern matrices

A factorization of an n×n nonnegative symmetric matrix A of the form BCBT, where C is a k×k symmetric matrix, and both B and C are required to be nonnegative, is called the Symmetric Nonnegative Matrix Trifactorization (SN-Trifactorization). The SNT-rank of A is the minimal k for which such factorization exists. The SNT-rank of a simple graph G that allows loops is defined to be the minimal possible SNT-rank of all symmetric nonnegative matrices whose zero-nonzero pattern is prescribed by the graph G.We define set-join covers of graphs, and show that finding the SNT-rank of G is equivalent to finding the minimal order of a set-join cover of G. Using this insight we develop basic properties of the SNT-rank for graphs and compute it for trees and cycles without loops. We show the equivalence between the SNT-rank for complete graphs and the Katona problem, and discuss uniqueness of patterns of matrices in the factorization.

Advances on similarity via transversal intersection of manifolds

Let A be an n×n real matrix. As shown in the recent paper S.M. Fallat, H.T. Hall, J.C.-H. Lin, and B.L. Shader (2022) [5], if the manifolds MA={G−1AG:G∈GL(n,R)} and Q(sgn(A)) (consisting of all real matrices having the same sign pattern as A), both considered as embedded submanifolds of Rn×n, intersect transversally at A, then every superpattern of sgn(A) also allows a matrix similar to A. Those authors introduced a condition on A (in terms of certain linear matrix equations) equivalent to the above transversality, called the nonsymmetric strong spectral property (nSSP). In this paper, this transversality property of A is characterized using an alternative, more direct and convenient condition, called the similarity-transversality property (STP). Let X=[xij] be a generic matrix of order n whose entries are independent variables. The STP of A is defined as the full row rank property of the Jacobian matrix of the entries of AX−XA at the zero entry positions of A with respect to the nondiagonal entries of X. This new approach makes it possible to take better advantage of the combinatorial structure of the matrix A, and provides theoretical foundation for constructing matrices similar to a given matrix while the entries have certain desired signs. In particular, several important classes of zero-nonzero patterns and sign patterns that require or allow this transversality property are identified. Examples illustrating many possible applications (such as diagonalizability, number of distinct eigenvalues, nilpotence, idempotence, semi-stability, eigenvalues and their algebraic and geometric multiplicities, Jordan canonical form, minimal polynomial, and rank) are provided.

Publisher's Note

Let A be an n×n real matrix. As shown in the recent paper S.M. Fallat, H.T. Hall, J.C.-H. Lin, and B.L. Shader (2022) [5], if the manifolds MA={G−1AG:G∈GL(n,R)} and Q(sgn(A)) (consisting of all real matrices having the same sign pattern as A), both considered as embedded submanifolds of Rn×n, intersect transversally at A, then every superpattern of sgn(A) also allows a matrix similar to A. Those authors introduced a condition on A (in terms of certain linear matrix equations) equivalent to the above transversality, called the nonsymmetric strong spectral property (nSSP). In this paper, this transversality property of A is characterized using an alternative, more direct and convenient condition, called the similarity-transversality property (STP). Let X=[xij] be a generic matrix of order n whose entries are independent variables. The STP of A is defined as the full row rank property of the Jacobian matrix of the entries of AX−XA at the zero entry positions of A with respect to the nondiagonal entries of X. This new approach makes it possible to take better advantage of the combinatorial structure of the matrix A, and provides theoretical foundation for constructing matrices similar to a given matrix while the entries have certain desired signs. In particular, several important classes of zero-nonzero patterns and sign patterns that require or allow this transversality property are identified. Examples illustrating many possible applications (such as diagonalizability, number of distinct eigenvalues, nilpotence, idempotence, semi-stability, eigenvalues and their algebraic and geometric multiplicities, Jordan canonical form, minimal polynomial, and rank) are provided.

Zero-nonzero tree patterns that allow

For , let . A zero-nonzero pattern of order n allows if , the inertia set of , and requires if . The study of zero-nonzero patterns allowing was introduced by Berliner et al. [Inertia sets allowed by matrix patterns. Electron J Linear Algebra. 2018;34:343–355]. In this paper, we give characterizations of irreducible zero-nonzero path patterns and double-star patterns that allow . Moreover, all irreducible zero-nonzero tree patterns of order 5 and 6 that allow are also characterized.

Algebraic conditions and the sparsity of spectrally arbitrary patterns

Abstract Given a square matrix A, replacing each of its nonzero entries with the symbol * gives its zero-nonzero pattern. Such a pattern is said to be spectrally arbitrary when it carries essentially no information about the eigenvalues of A. A longstanding open question concerns the smallest possible number of nonzero entries in an n × n spectrally arbitrary pattern. The Generalized 2n Conjecture states that, for a pattern that meets an appropriate irreducibility condition, this number is 2n. An example of Shitov shows that this irreducibility is essential; following his technique, we construct a smaller such example. We then develop an appropriate algebraic condition and apply it computationally to show that, for n ≤ 7, the conjecture does hold for ℝ, and that there are essentially only two possible counterexamples over ℂ. Examining these two patterns, we highlight the problem of determining whether or not either is in fact spectrally arbitrary over ℂ. A general method for making this determination for a pattern remains a major goal; we introduce an algebraic tool that may be helpful.

Zero-nonzero Patterns that Allow or Require an Inertia Set Related to Dynamical Systems

The inertia of an $n\times n$ real matrix $B$, denoted by $\hbox{i}(B)$, is the ordered triple $\hbox{i}(B)=(i_+(B)$, $i_-(B), i_0(B) )$, in which $i_+(B)$, $i_-(B)$ and $i_0(B)$ are the numbers of its eigenvalues (counting multiplicities) with positive, negative and zero real parts, respectively. The inertia of an $n\times n$ zero-nonzero pattern ${\cal A}$ is the set $\hbox{i}({\cal A})=\{\hbox{i}(B) \mid B\in Q({\cal A})\}$. For $n\ge 2$, let $$\mathbb{S}_n^*=\{(0,n,0), (0,n-1,1), (1,n-1,0), (n,0,0), (n-1,0,1), (n-1,1,0)\}.$$
 An $n\times n$ zero-nonzero pattern ${\cal A}$ allows $\mathbb{S}_n^*$ if $\mathbb{S}_n^* \subseteq \hbox{i}({\cal A})$ and requires $\mathbb{S}_n^*$ if $\mathbb{S}_n^*=\hbox{i}({\cal A})$. In this paper, it is shown that there are no zero-nonzero patterns for order $n\ge 2$ that require $\mathbb{S}_n^*$. Also, a complete characterization of zero-nonzero star patterns of order $n\ge 3$ that allow $\mathbb{S}_n^*$ is given.

The relationship between triangle size and minimum rank for 7 × 7 zero-nonzero patterns

In this paper, a positive answer to the question (I) in Johnson and Link [C. R. Johnson, J. A. Link, The extent to which triangular sub-patterns explain minimum rank, Discrete Applied Mathematics, 156 (2008) 1637-1651] is given for 7 × 7 zero-nonzero patterns.

Spectrally arbitrary patterns over rings with unity

A zero-nonzero pattern A is a matrix with entries from the set {0,⁎}. A square pattern A is spectrally arbitrary over R, a commutative ring with unity, if for each n-th degree monic polynomial f(x)∈R[x], there exists a matrix A over R with pattern A such that the characteristic polynomial pA(x)=f(x). A pattern A is relaxed spectrally arbitrary over R if for each n-th degree monic polynomial f(x)∈R[x], there exists a matrix A over R with either pattern A or a subpattern of A such that the characteristic polynomial pA(x)=f(x). We evaluate how the structure of rings affects how we determine if a pattern is spectrally arbitrary. We consider whether a pattern A that is spectrally arbitrary over R is spectrally arbitrary or relaxed spectrally arbitrary over another commutative ring S with unity. In particular, we establish that a pattern that is spectrally arbitrary over Z is relaxed spectrally arbitrary over Zm for all m∈Z+ and spectrally arbitrary over Q. Similarly, a pattern that is spectrally arbitrary over the p-adic integers is spectrally arbitrary over the p-adic numbers.

Zero-nonzero patterns of order n ≥ 4 do not require [formula omitted

In [1], the set of refined inertias Hn for sign patterns is expanded to Hn⁎ for zero-nonzero patterns. Also, it is proved that there are no zero-nonzero patterns of even order 2m that require H2m⁎. In this paper, it is shown that there are no zero-nonzero patterns of odd order n≥5 that require Hn⁎, which together with the previous result prove that there are no zero-nonzero patterns of order n≥4 that require Hn⁎.

Inertia sets allowed by matrix patterns

Motivated by the possible onset of instability in dynamical systems associated with a zero eigenvalue, sets of inertias $\sn_n$ and $\SN{n}$ for sign and zero-nonzero patterns, respectively, are introduced. For an $n\times n$ sign pattern $\mc{A}$ that allows inertia $(0,n-1,1)$, a sufficient condition is given for $\mc{A}$ and every superpattern of $\mc{A}$ to allow $\sn_n$, and a family of such irreducible sign patterns for all $n\geq 3$ is specified. All zero-nonzero patterns (up to equivalence) that allow $\SN{3}$ and $\SN{4}$ are determined, and are described by their associated digraphs.

The ray nonsingularity of certain uniformly random ray patterns

A uniformly random ray pattern matrix A with a given zero-nonzero pattern (described by a digraph D with no multi-arcs or loops) is the matrix whose nonzero entries are mutually independent random variables uniformly distributed over the unit circle S1 in the complex plane. It is shown in this paper that the probability of I−A to be ray nonsingular is completely determined by the cycle graph CG(D) of D (i.e. the adjacency structure of the directed cycles in D) if CG(D) is a tree. A formula is given to compute the probability when CG(D) is a tree, and it is also shown that as the order of CG(D) tends to infinity, the limit of the probability is 0.

Sets of refined inertias of zero–nonzero patterns

A set Hn⁎ of refined inertias for zero–nonzero patterns is introduced that is analogous to the set Hn previously considered for sign patterns. For n=3 and 4, a complete characterization of irreducible zero–nonzero patterns that allow or require Hn⁎ is given, and each zero–nonzero pattern that allows Hn⁎ has a signing that allows Hn. In contrast, for n≥5 a family of irreducible zero–nonzero patterns is given that allows Hn⁎ but for which no one signing allows Hn.

The minimum rank problem for circulants

The minimum rank problem is to determine for a graph G the smallest rank of a Hermitian (or real symmetric) matrix whose off-diagonal zero-nonzero pattern is that of the adjacency matrix of G. Here G is taken to be a circulant graph, and only circulant matrices are considered. The resulting graph parameter is termed the minimum circulant rank of the graph. This value is determined for every circulant graph in which a vertex neighborhood forms a consecutive set, and in this case is shown to coincide with the usual minimum rank. Under the additional restriction to positive semidefinite matrices, the resulting parameter is shown to be equal to the smallest number of dimensions in which the graph has an orthogonal representation with a certain symmetry property, and also to the smallest number of terms appearing among a certain family of polynomials determined by the graph. This value is then determined when the number of vertices is prime. The analogous parameter over R is also investigated.

Integrally normalizable matrices and zero–nonzero patterns

The problem of determining whether or not an n×n integer matrix is diagonally similar to an integer matrix with n−1 off-diagonal entries equal to 1 is studied. Such a matrix is called integrally normalizable, and a zero–nonzero pattern is integrally normalizable if each matrix with this zero–nonzero pattern is integrally normalizable with respect to the same set of n−1 off-diagonal entries. Matrices that are integrally normalizable with respect to a fixed spanning tree, and integrally normalizable zero–nonzero patterns are characterized. The maximum number of nonzero entries in an n×n integrally normalizable zero–nonzero pattern is shown to be n2+3n−22. Extensions to matrices over other integral domains are also presented.

Zero forcing for sign patterns

We introduce a new variant of zero forcing—signed zero forcing. The classical zero forcing number provides an upper bound on the maximum nullity of a matrix with a given graph (i.e. zero-nonzero pattern). Our new variant provides an analogous bound for the maximum nullity of a matrix with a given sign pattern. This allows us to compute, for instance, the maximum nullity of a Z-matrix whose graph is L(Kn), the line graph of a clique.

Inverse invariant zero–nonzero patterns

We determine the irreducible zero–nonzero patterns A such that for any nonsingular matrix B over a field with zero–nonzero pattern A, the inverse B−1 has the same zero–nonzero pattern A. We also determine the zero–nonzero patterns P such that for any nonsingular matrix Q over a field with zero–nonzero pattern P, the transpose of the inverse (Q−1)T has the same zero–nonzero pattern P. One application of these results is to deduce the corresponding results on sign patterns.

Monomials in quadratic forms

We obtain some constraints on the zero-nonzero pattern of entries in the matrix of a real quadratic form which attains a minimum on a large set of vertices in the multidimensional cube centered at the origin whose edges are parallel to the coordinate axes. In particular, if the graph of the matrix contains an articulation point then the set of the minima of the corresponding quadratic form is not maximal (with respect to set inclusion) among all such sets for various quadratic forms.

Minimum rank, maximum nullity, and zero forcing number of simple digraphs

A simple digraph describes the off-diagonal zero-nonzero pattern of a family of (not necessarily symmetric) matrices. Minimum rank of a simple digraph is the minimum rank of this family of matrices; maximum nullity is defined analogously. The simple digraph zero forcing number is an upper bound for maximum nullity. Cut-vertex reduction formulas for minimum rank and zero forcing number for simple digraphs are established. The effect of deletion of a vertex on minimum rank or zero forcing number is analyzed, and simple digraphs having very low or very high zero forcing number are characterized.

Bounds for minimum semidefinite rank from superpositions and cutsets

The real (complex) minimum semidefinite rank of a graph is the minimum rank among all real symmetric (complex Hermitian) positive semidefinite matrices that are naturally associated via their zero-nonzero pattern to the given graph. In this paper we give an upper bound on the minimum semidefinite rank of a graph when the graph is modified from the superposition of two graphs by cancelling some number of edges. We also provide a lower bound for the minimum semidefinite rank of a graph determined by a given cutset. When the complement of the cutset is a star forest these lower and upper bounds coincide and we can compute the minimum semidefinite rank in terms of smaller graphs. This result encompasses the previously known case in which the cut set has order two or smaller. Next we provide results for when the cut set has order three. Using these results we provide an example where the positive semidefinite zero forcing number is strictly greater than the maximum positive semidefinite nullity.

A new lower bound for the positive semidefinite minimum rank of a graph

The real positive semidefinite minimum rank of a graph is the minimum rank among all real positive semidefinite matrices that are naturally associated via their zero-nonzero pattern to the given graph. In this paper, we use orthogonal vertex removal and sign patterns to improve the lower bound for the real positive semidefinite minimum rank determined by the OS-number and the positive semidefinite zero forcing number.