On a finite rational criterion for the irreducibility of a matrix

A matrix A ∈ M _n (C) is said to be irreducible if the only orthoprojectors that commute with A are the zero and unit matrices. A finite rational criterion for irreducibility is proposed. The criteria for verification of this property that can be found in the literature are neither finite nor rational.     

Graph-theoretic characterization of the matrix property of full irreducibility without using a transversal

In this paper we introduce the notion of strongly connected polygons in the line graph of a bipartite graph. We use this notion to give a necessary and sufficient condition for a matrix Y to be fully irreducible without the need to construct a transversal of Y. In addition, we show that the notion of strongly connected polygons forms the basis of a general theory that may be used for finding all the cycles in the directed graph of a fully irreducible matrix and for constructing all the nonzero transversals of a fully irreducible matrix.     

A sufficient condition for a non-negative matrix to have non-negative determinant

It is shown that a matrix with non-negative entries has non-negative determinant if in each row the elements decrease, by steadily smaller amounts, as one proceeds (in either direction) away from the main diagonal. This condition suffices to establish non- negativity of the determinant for certain matrices to which the familiar Minkowski- Hadamard-Ostrowski dominance conditions do not apply. In the symmetric case it provides a sufficient condition for non-negative definiteness. This may be applied to establish the positive definiteness of certain real symmetric Toeplitz matrices.     

Dimension group of a non-negative integer irreducible matrix

In this paper we are concerned with a non-negative integer and irreducible matrix $A:{\mathbb{Z}}^d\to {\mathbb{Z}}^d$. The main contribution is to prove that if the matrix satisfies certain spectral and algebraic constraints, the cone:$C=\{v\in {\mathbb{Z}}^d/\exists n⩾0\mathrm{and}{A}^{n}v⩾0\}\subset {\mathbb{Z}}^d$is defined by linear maps ${\varphi }_0\text{,}\dots \text{,}{\varphi }_{k-1}:{\mathbb{Z}}^d\to \mathbb{R}$, in the sense that v ∈ C is equivalent to, ϕ_l(v) ⩾ 0 for all l = 0, … , k − 1 (where k is the index of cyclicity of the irreducible matrix). This result allows us to characterize the dimension group generated by the matrix, it is a subgroup of ${\mathbb{R}}^k$ endowed with an order induced by the positive cone of ${\mathbb{R}}^k$.     

Complex eigenvalues of a non-negative matrix with a specified graph

Let A be a non-negative matrix of order n with Perron eigenvalue ? and associated directed graph G. Let m be the length of the longest circuit of G. Theorem: If m=2, all eigenvalues of A are real. If 2<m?n, and if λ=μ+iv is an eigenvalue of A, then $\text{μ+|v|tan(}\text{Π}\text{m}\text{) ? ρ}$.     

On a characterization of complete matrix rings

Peter R. Fuchs established in 1991 a new characterization of complete matrix rings by showing that a ringR with identity is isomorphic to a matrix ringM _n (S) for some ringS (and somen 2) if and only if there are elementsx andy inR such thatx ^n–1 0,x ⁿ=0=y ²,x+y is invertible, and Ann(x ^n–1)Ry={0} (theintersection condition), and he showed that the intersection condition is superfluous in casen=2. We show that the intersection condition cannot be omitted from Fuchs' characterization ifn3; in fact, we show that if the intersection condition is omitted, then not only may it happen that we do not obtain a completen ×n matrix ring for then under consideration, but it may even happen that we do not obtain a completem ×m matrix ring for anym2.     

On the irreducibility of multivariate subresultants

Let P₁,…,P_n be generic homogeneous polynomials in n variables of degrees d₁,…,d_n respectively. We prove that if ν is an integer satisfying ∑_i=1ⁿd_i?n+1?min{d_i}<ν, then all multivariate subresultants associated to the family P₁,…,P_n in degree ν are irreducible. We show that the lower bound is sharp. As a byproduct, we get a formula for computing the residual resultant of $\text{ρ?ν+n?1}\text{n?1}$ smooth isolated points in $\text{P}{}^{\mathrm{<mspace\; sp="0.2"\; width="2px"\; linebreak="nobreak"\; is="true"></mspace><mtext>n?1</mtext>}}\text{.}$To cite this article: L. Busé, C. D'Andrea, C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

On the irreducibility of Hecke polynomials

Let be the characteristic polynomial of the th Hecke operator acting on the space of cusp forms of weight for the full modular group. We record a simple criterion which can be used to check the irreducibility of the polynomials . Using this criterion with some machine computation, we show that if there exists such that is irreducible and has the full symmetric group as Galois group, then the same is true of for each prime .

     

On local irreducibility of the spectrum

Let be the algebra of complex matrices, and for denote by and the spectrum and spectral radius of respectively. Let be a domain in containing 0, and let be a holomorphic map. We prove: (1) if for , then for ; (2) if for , then again for . Both results are special cases of theorems expressing the irreducibility of the spectrum near .

     

On the irreducibility of a certain class of Laguerre polynomials

R. Gow has investigated the problem of determining classical polynomials with Galois group A_m, the alternating group on m letters, in the case that m is even (odd m being previously handled in work of I. Schur). He showed that the generalized Laguerre polynomial L_m^(m)(x), defined below, has Galois group A_m provided m>2 is even and L_m^(m)(x) is irreducible (and obtained irreducibility in some cases). In this paper, we establish that L_m^(m)(x) is irreducible for almost all m (and, hence, has Galois group A_m for almost all even m).     

On the irreducibility of a class of polynomials,III

This work is a continuation and extension of our earlier articles on irreducible polynomials. We investigate the irreducibility of polynomials of the form g(f(x)) over an arbitrary but fixed totally real algebraic number field $\text{L}$, where g(x) and f(x) are monic polynomials with integer coefficients in $\text{L}$, g is irreducible over $\text{L}$ and its splitting field is a totally imaginary quadratic extension of a totally real number field. A consequence of our main result is as follows. If g is fixed then, apart from certain exceptions f of bounded degree, g(f(x)) is irreducible over $\text{L}$ for all f having distinct roots in a given totally real number field.     

Inverse iteration for calculating the spectral radius of a non-negative irreducible matrix

Noda established the superlinear convergence of an inverse iteration procedure for calculating the spectral radius and the associated positive eigenvector of a non-negative irreducible matrix. Here a new proof is given, based completely on the underlying order structure. The main tool is Hopf's inequality. It is shown that the convergence is quadratic.     

On a transport problem and monoids of non-negative integers

A problem about how to transport profitably a group of cars leads us to studying the set T formed by the integers n such that the system of inequalities, with non-negative integer coefficients,

$$\begin{aligned} a_1x_1 +\cdots + a_px_p + \alpha \le n \le b_1x_1 +\cdots + b_px_p - \beta \end{aligned}$$

has at least one solution in \({\mathbb N}^p\). We prove that \(T\cup \{0\}\) is a submonoid of \(({\mathbb N},+)\) and, moreover, we give algorithmic processes to compute T.

     

Note on the computation of the maximal eigenvalue of a non-negative irreducible matrix

Summary We prove the convergence of Wielandt's method in the computation of the maximal eigenvalue and eigenvector of a non-negative irreducible matrix.