Invertible and nilpotent matrices over antirings

In this paper, we characterize invertible matrices over an arbitrary commutative antiring S with 1 and find the structure of GL_n(S). We find the number of nilpotent matrices over an entire commutative finite antiring. We prove that every nilpotent n×n matrix over an entire antiring can be written as a sum of ⌈log₂n⌉ square-zero matrices and also find the necessary number of square-zero summands for an arbitrary trace-zero matrix to be expressible as their sum.     

Noncommuting graphs of matrices over semirings

We study diameters and girths of noncommuting graphs of semirings. For a noncommutative semiring that is either multiplicatively or additively cancellative, we find the diameter and the girth of its noncommuting graph and prove that it is Hamiltonian. Moreover, we find diameters and girths of noncommuting graphs of all nilpotent matrices over a semiring, all invertible matrices over a semiring, all noninvertible matrices over a semiring, and the full matrix semiring. In nearly all cases we prove that diameters are less than or equal to 2 and girths are less than or equal to 3, except in the case of 2×2 nilpotent matrices.     

Commuting graphs of matrices over semirings

We calculate diameters and girths of commuting graphs of the set of all nilpotent matrices over a semiring, the group of all invertible matrices over a semiring, and the full matrix semiring.     

Bijective linear rank preservers for spaces of matrices over antinegative semirings

We classify the bijective linear operators on spaces of matrices over antinegative commutative semirings with no zero divisors which preserve certain rank functions such as the symmetric rank, the factor rank and the tropical rank. We also classify the bijective linear operators on spaces of matrices over the max-plus semiring which preserve the Gondran-Minoux row rank or the Gondran-Minoux column rank.     

On nilpotent subsemigroups of the matrix semigroup over an antiring

In this paper, nilpotent subsemigroups in the matrix semigroup over a commutative antiring are discussed. Some basic properties and characterizations for the nilpotent subsemigroups are given, and some equivalent conditions for the matrix semigroup over a commutative antiring to have a maximal nilpotent subsemigroup are obtained. Also, the maximal nilpotent subsemigroups in the matrix semigroup are described.     

Positive definiteness of Hermitian interval matrices

We present a new necessary and sufficient criterion to check the positive definiteness of Hermitian interval matrices. It is shown that an n×n Hermitian interval matrix is positive definite if and only if its 4^n-1(n-1)! specially chosen Hermitian vertex matrices are positive definite.     

Singular points of cyclic weighted shift matrices

Let S be an n-by-n   cyclic weighted shift matrix, and _FS(t,x,y)=det(tI+xℜ(S)+yℑ(S))$F_S(t,x,y)=\det (tI+x\Re (S)+y\Im (S))$ be a ternary form associated with S  . We investigate the number of singular points of the curve _FS(t,x,y)=0$F_S(t,x,y)=0$, and show that the number of singular points of _FS(t,x,y)=0$F_S(t,x,y)=0$ associated with a cyclic weighted shift matrix whose weights are neither 1-periodic nor 2-periodic is less than or equal to n(n−3)/2$n(n-3)/2$. Furthermore, we verify the upper bound n(n−3)/2$n(n-3)/2$ is sharp for 4?n?7$4?n?7$.     

Fixed zeros in the model matching problem for systems over semirings

This paper studies “fixed zeros” of solutions to the model matching problem for systems over semirings. Such systems have been used to model queueing systems, communication networks, and manufacturing systems. The main contribution of this paper is the discovery of two fixed zero structures, which possess a connection with the extended zero semimodules of solutions to the model matching problem. Intuitively, the fixed zeros provides an essential component that is obtained from the solutions to the model matching problem. For discrete event dynamic systems modeled in max-plus algebra, a common Petri net component constructed from the solutions to the model matching problem can be discovered from the fixed zero structure.     

Inequalities for Gondran-Minoux rank and idempotent semirings

The rank-sum, rank-product, and rank-union inequalities for Gondran-Minoux rank of matrices over idempotent semirings are considered. We prove these inequalities for matrices over quasi-selective semirings without zero divisors, which include matrices over the max-plus semiring. Moreover, it is shown that the inequalities provide the linear algebraic characterization for the class of quasi-selective semirings. Namely, it is proven that the inequalities hold for matrices over an idempotent semiring S without zero divisors if and only if S is quasi-selective. For any idempotent semiring which is not quasi-selective it is shown that the rank-sum, rank-product, and rank-union inequalities do not hold in general. Also, we provide an example of a selective semiring with zero divisors such that the rank-sum, rank-product, and rank-union inequalities do not hold in general.     

Numerical ranges of weighted shift matrices

We show that if A is an n-by-n (n?3) matrix of the form

     

Numerical ranges of reducible companion matrices

In this paper, we show that a reducible companion matrix is completely determined by its numerical range, that is, if two reducible companion matrices have the same numerical range, then they must equal to each other. We also obtain a criterion for a reducible companion matrix to have an elliptic numerical range, put more precisely, we show that the numerical range of an n-by-n reducible companion matrix C is an elliptic disc if and only if C is unitarily equivalent to A⊕B, where A∈M_n-2, B∈M₂ with σ(B)={aω₁,aω₂}, , ω₁≠ω₂, and .     

Some inequalities for communtators and an application to spectral variation

Summary If –I is a positive semidefinite operator andA andB are either both Hermitian or both unitary, then every unitarily invariant norm ofA–B is shown to be bounded by that ofA–B. Some related inequalities are proved. An application leads to a generalization of the Lidskii-Wielandt inequality to matrices similar to Hermitian.Dedicated to the memory of Alexander M. Ostrowski on the occasion of the 100th anniversary of his birth     

Relative perturbation results for eigenvalues and eigenvectors of diagonalisable matrices

Let and be a perturbed eigenpair of a diagonalisable matrixA. The problem is to bound the error in and . We present one absolute perturbation bound and two relative perturbation bounds. The absolute perturbation bound is an extension of Davis and Kahan's sin θ Theorem from Hermitian to diagonalisable matrices. The two relative perturbation bounds assume that and are an exact eigenpair of a perturbed matrixD ₁ AD ₂ , whereD ₁ andD ₂ are non-singular, butD ₁ AD ₂ is not necessarily diagonalisable. We derive a bound on the relative error in and a sin θ theorem based on a relative eigenvalue separation. The perturbation bounds contain both the deviation ofD ₁ andD ₂ from similarity and the deviation ofD ₂ from identity. This work was partially supported by NSF grant CCR-9400921.     

On linear combinations of two idempotent matrices over an arbitrary field

Given an arbitrary field K and non-zero scalars α and β, we give necessary and sufficient conditions for a matrix A∈M_n(K) to be a linear combination of two idempotents with coefficients α and β. This extends results previously obtained by Hartwig and Putcha in two ways: the field K considered here is arbitrary (possibly of characteristic 2), and the case α≠±β is taken into account.     

Polynomial numerical hulls of some normal matrices

In this note, polynomial numerical hulls of matrices of the form _A1⊕i_A2$A_1\oplus i{A}_2$, where _A1$A_1$ and _A2$A_2$ are Hermitian, are characterized.     

Spectral variation bounds for diagonalisable matrices

Summary This note is related to an earlier paper by Bhatia, Davis, and Kittaneh [4]. For matrices similar to Hermitian, we prove an inequality complementary to the one proved in [4, Theorem 3]. We also disprove a conjecture made in [4] about the norm of a commutator. This work was done when the first author visited the SFB 343 at University of Bielefeld in May and June 1994.     

Rank-deficient submatrices of Fourier matrices

We consider the maximal rank-deficient submatrices of Fourier matrices with order a power of a prime number. We do this by considering a hierarchical subdivision of these matrices into low rank blocks. We also explore some connections with the fast Fourier transform (FFT), and with an uncertainty principle for Fourier transforms over finite Abelian groups.     

On approximately simultaneously diagonalizable matrices

A collection A₁, A₂, …, A_k of n × n matrices over the complex numbers C has the ASD property if the matrices can be perturbed by an arbitrarily small amount so that they become simultaneously diagonalizable. Such a collection must perforce be commuting. We show by a direct matrix proof that the ASD property holds for three commuting matrices when one of them is 2-regular (dimension of eigenspaces is at most 2). Corollaries include results of Gerstenhaber and Neubauer-Sethuraman on bounds for the dimension of the algebra generated by A₁, A₂, …, A_k. Even when the ASD property fails, our techniques can produce a good bound on the dimension of this subalgebra. For example, we establish for commuting matrices A₁, …, A_k when one of them is 2-regular. This bound is sharp. One offshoot of our work is the introduction of a new canonical form, the H-form, for matrices over an algebraically closed field. The H-form of a matrix is a sparse “Jordan like” upper triangular matrix which allows us to assume that any commuting matrices are also upper triangular. (The Jordan form itself does not accommodate this.)     

Eigenvalues of large sample covariance matrices of spiked population models

We consider a spiked population model, proposed by Johnstone, in which all the population eigenvalues are one except for a few fixed eigenvalues. The question is to determine how the sample eigenvalues depend on the non-unit population ones when both sample size and population size become large. This paper completely determines the almost sure limits of the sample eigenvalues in a spiked model for a general class of samples.