For matrices with all nonnegative entries, the Perron-Frobenius theorem guarantees the existence of an eigenvector with all nonnegative components. We show that the existence of such an eigenvector is also guaranteed for a very different class of matrices, namely real symmetric matrices with exactly two eigenvalues. We also prove a partial converse, that among real symmetric matrices with any more than two eigenvalues there exist some having no nonnegative eigenvector.
We prove the existence of a common eigenvector for commutative, nilpotent and quasinilpotent semigroups of matrices with complex or real nonnegative entries both in the conventional and tropical linear algebra.
In this paper, we show that all totally nonnegative GCD matrices are always divisors of the corresponding LCM matrices in the ring $\mathbb{M}_{n}(\mathbb{Z})$. We also introduce \lq\lq column monotone matrices" used to construct all totally nonegative GCD matrices.
We establish a factorisation theorem for invertible, cross-symmetric, totally nonnegative matrices, and illustrate the theory by verifying that certain cases of Holte's Amazing Matrix are totally nonnegative.
The problem of determining necessary and sufficient conditions for a set of real numbers to be the eigenvalues of a symmetric nonnegative matrix is called the symmetric nonnegative inverse eigenvalue problem (SNIEP). In this paper we solve SNIEP in the case of trace zero symmetric nonnegative 5x5 matrices.
Given a finite set $\{M_0,\dots,M_{d-1}\}$ of nonnegative $2\times 2$ matrices and a nonnegative column-vector $V$, we associate to each $(ω_n)\in\{0,\dots,d-1\}^\mathbb N$ the sequence of the column-vectors $\displaystyle{M_{ω_1}\dots M_{ω_n}V\over\Vert M_{ω_1}\dots M_{ω_n}V\Vert}$. We give the necessary and sufficient condition on the matrices $M_k$ and the vector $V$ for this sequence to converge for all \hbox{$(ω_n)\in\{0,\dots,d-1\}^\mathbb N$} such that $\forall n,\ M_{ω_1}\dots M_{ω_n}V\ne\begin{pmatrix}0\\0\end{pmatrix}$.
We consider polynomials in R[x] which map the set of nonnegative (element-wise) matrices of a given order into itself. Let n be a positive integer and define P(n)= {p in R[x] : p(A) is nonnegative (element-wise), for all A, A an n-by-n nonnegative (element-wise) matrix}. This set plays a role in the Nonnegative Inverse Eigenvalue Problem. Clark and Paparella conjectured that P(n+1) is strictly contained in P(n). We prove this conjecture.
Using matrix function theory, Perron-Frobenius theory, combinatorial matrix theory, and elementary number theory, we characterize, classify, and describe in terms of the Jordan canonical form the matrix pth-roots of imprimitive irreducible nonnegative matrices. Preliminary results concerning the matrix roots of reducible matrices are provided as well.
In this paper, we prove that the semigroups of invertible matrices with nonnegative elements over linearly oredered associative rings are elementarily equivalent if and only if the matrices have the same dimension and the rings are elementarily equivalent as ordered rings.
Using the relationship between totally nonnegative matrices and directed acyclic weighted planar networks, we show that $2\times 2$ minors of minor matrices of totally nonnegative matrices are also nonnegative. We give a combinatorial interpretation for the minors of minor matrices in terms of the weights of families of paths in a network.
A known characterization for entire functions that preserve all nonnegative matrices of order two is shown to characterize polynomials that preserve nonnegative matrices of order two. Equivalent conditions are derived and used to prove that $\mathscr{P}_3 \subset \mathscr{P}_2$, which was previously unknown. A new characterization is given for polynomials that preserve nonnegative circulant matrices of order two.
Denton, Parke, Tao and Zhang gave a new method which determines eigenvectors from eigenvalues for Hermitian matrices with distinct eigenvalues. In this short note, we extend the above result to general Hermitian matrices.
It is known that the critical exponent (CE) for conventional, continuous powers of $n$-by-$n$ doubly nonnegative (DN) matrices is $n-2$. Here, we consider the larger class of diagonalizable, entry-wise nonnegative $n$-by-$n$ matrices with nonnegative eigenvalues (GDN). We show that, again, a CE exists and are able to bound it with a low-coefficient quadratic. However, the CE is larger than in the DN case; in particular, 2 for $n=3$. There seems to be a connection with the index of primitivity, and a number of other observations are made and questions raised. It is shown that there is no CE for…
In this paper we present some symmetric nonnegative 5x5 matrix families that realize a previously unknown region. We also prove that these and other symmetric nonnegative 5x5 matrix families are closed under perturbations first presented in W. Guo. Eigenvalues of nonnegative matrices. Linear Algebra and its Applications, 266:261-270, 1997.
A matrix is $k$-nonnegative if all its minors of size $k$ or less are nonnegative. We give a parametrized set of generators and relations for the semigroup of $k$-nonnegative $n\times n$ invertible matrices in two special cases: when $k = n-1$ and when $k = n-2$, restricted to unitriangular matrices. For these two cases, we prove that the set of $k$-nonnegative matrices can be partitioned into cells based on their factorizations into generators, generalizing the notion of Bruhat cells from totally nonnegative matrices. Like Bruhat cells, these cells are homeomorphic to open balls and have a to…
We prove that there exists an exponent beyond which all continuous conventional powers of n-by-n doubly nonnegative matrices are doubly nonnegative. We show that this critical exponent cannot be less than $n-2$ and we conjecture that it is always $n-2$ (as it is with Hadamard powering). We prove this conjecture when $n<6$ and in certain other special cases. We establish a quadratic bound for the critical exponent in general.
In this paper we prove that every automorphism of the semigroup of invertible matrices with nonnegative elements over a linearly oredered associative ring on some specially defined subgroup concides with the composition of an inner automorphism of the semigroup, an automorphism of the ring preserving the order, and a central homothety.
Algebras generated by strictly positive matrices are described up to similarity, including the commutative, simple, and semisimple cases. We provide sufficient conditions for some block diagonal matrix algebras to be generated by a set of nonnegative matrices up to similarity. Also we find all realizable dimensions of algebras generated by two nonnegative semi-commuting matrices. The last result provides the solution to the problem posed by M. Kandić, K. Šivic (2017).
In this paper, we answer the various forms of nonnegative inverse eigenvalue problems with prescribed diagonal entries for order three: real or complex general matrices, symmetric stochastic matrices, and real or complex doubly stochastic matrices. We include the known cases, the symmetric matrices and real or complex stochastic matrices, to compare the other results and for completeness. In addition, for a given list of eigenvalues, we compute the exact range for the largest value of the diagonal entries of the various nonnegative matrices.
This paper studies commuting matrices in max algebra and nonnegative linear algebra. Our starting point is the existence of a common eigenvector, which directly leads to max analogues of some classical results for complex matrices. We also investigate Frobenius normal forms of commuting matrices, particularly when the Perron roots of the components are distinct. For the case of max algebra, we show how the intersection of eigencones of commuting matrices can be described, and we consider connections with Boolean algebra which enables us to prove that two commuting irreducible matrices in max a…
Riordan matrices are infinite lower triangular matrices determined by a pair of formal power series over the real or complex field. These matrices have been mainly studied as combinatorial objects with an emphasis placed on the algebraic or combinatorial structure. The present paper contributes to the linear algebraic discussion with an analysis of Riordan matrices by means of the interaction of the properties of formal power series with the linear algebra. Specifically, it is shown that if a Riordan matrix $A$ is an $n\times n$ pseudo-involution then the singular values of $A$ must come in re…
We study ranks of the $r\textrm{th}$ Hadamard powers of doubly nonnegative matrices and show that the matrix $A^{\circ r}$ is positive definite for every $n\times n$ doubly nonnegative matrix $A$ and for every $r>n-2$ if and only if no column of $A$ is a scalar multiple of any other column of $A.$ A particular emphasis is given to the study of rank, positivity and monotonicity of Hadamard powers of rank two, positive semidefinite matrices that have all entries positive.
Roots of matrices are well-studied. The conditions for their existence are understood: The block sizes of nilpotent Jordan blocks, arranged in pairs, have to satisfy some simple algebraic property. More interesting are structured roots of structured matrices. Probably the best known example is the existence and uniqueness of positive definite square roots of a positive definite matrix. If one drops the requirement of positive definiteness of the square root, it turns out that there exists an abundance of square roots. Here a description of all canonical forms of all square roots is possible an…
In this article the well known "Perron-Frobenius theory" is investigated involving the higher rank numerical range $Λ_{k}(A)$ of an irreducible and entrywise nonnegative matrix $A$ and extending the notion of elements of maximum modulus in $Λ_{k}(A)$. Further, an application of this theory to the $Λ_{k}(L(λ))$ of a Perron polynomial $L(λ)$ is elaborated via its companion matrix $C_{L}$.
Suppose that R is an ordered ring, G_n(R) is a subsemigroup of $GL_n(R)$, consisting of all matrices with nonnegative elements. A.V. Mikhalev and M.A. Shatalova described all automorphisms of G_n(R), where R is a linearly ordered skewfield and n>1. E.I. Bunina and A.V. Mikhalev found all automorphisms of G_n(R), if R is an arbitrary linearly ordered associative ring with 1/2, n>2. In this paper we describe automorphisms of G_n(R), if R is a commutative partially ordered ring, containing Q, n>2.
Let $λ_1 \geq λ_2 \geq λ_3 \geq λ_4 \geq λ_5 \geq -λ_1$ be real numbers such that $\sum_{i=1}^5 λ_i =0$. In \cite{oren}, O. Spector prove that a necessary and sufficient condition for $λ_1, λ_2, λ_3, λ_4, λ_5$ to be the eigenvalues of a symmetric nonnegative $5 \times 5$ matrix is "$λ_2+λ_5<0$ and $\sum_{i=1}^5 λ_{i}^{3} \geq 0"$. In this article, we show that this condition is also a necessary and sufficient condition for $λ_1, λ_2, λ_3, λ_4, λ_5$ to be the spectrum of a traceless bisymmetric nonnegative $5 \times 5$ matrix.
If a left-product $M_n... M_1$ of square complex matrices converges to a nonnull limit when $n\to\infty$ and if the $M_n$ belong to a finite set, it is clear that there exists an integer $n_0$ such that the $M_n$, $n\ge n_0$, have a common right-eigenvector $V$ for the eigenvalue 1. Now suppose that the $M_n$ are nonnegative and that $V$ has positive entries. Denoting by $Δ$ the diagonal matrix whose diagonal entries are the entries of $V$, the stochastic matrices $S_n=Δ^{-1}M_nΔ$ satisfy $M_n... M_{n_0}=ΔS_n... S_{n_0}Δ^{-1}$, so the problem of the convergence of $M_n... M_1$ reduces to the o…
