Furtado, Susana Johnson, Charles R.
Published in
Linear Algebra and Its Applications

It is shown that for any k∈{1,2,…,(m−1)!} there exist m invertible complex matrices such that among the m! products Aσ=Aσ(1)Aσ(2)⋯Aσ(m), σ∈Sm, exactly k different similarity classes occur. The cases in which the matrices Ai are upper triangular or are 2-by-2 are considered in detail. In the former case, it is shown that any m−1 unispectral matrices...

Sato, Iwao
Published in
Linear Algebra and Its Applications

Recently, Bapat and Sivasubramanian [1] presented formulas for the determinant and the inverse of the product distance matrix of a tree. Furthermore, they defined a bivariant Ihara–Selberg zeta function of a graph and gave its determinant expression. We present new proofs for three results of Bapat and Sivasubramanian.

Beasley, LeRoy B. Mousley, Sarah
Published in
Linear Algebra and Its Applications

Let D be a directed graph (digraph) on n vertices. The digraph D is said to be primitive if for some m, between any ordered pair of vertices of D there is a directed walk of length m from the first vertex to the other. Here our focus is a generalization of primitivity, called k-primitivity, where k-arc-colorings of digraphs are considered. Let kmax...

Elton, Daniel M. Levitin, Michael Polterovich, Iosif
Published in
Annales Henri Poincaré

We study the spectrum of a one-dimensional Dirac operator pencil, with a coupling constant in front of the potential considered as the spectral parameter. Motivated by recent investigations of graphene waveguides, we focus on the values of the coupling constant for which the kernel of the Dirac operator contains a non-trivial square integrable func...

Trefois, Maguy Van Dooren, Paul Delvenne, Jean-Charles
Published in
Linear Algebra and Its Applications

In this paper, we consider the problem of factorizing the n×n matrix Jn of all ones into the n×n binary matrices. We show that under some conditions on the factors, these are isomorphic to a row permutation of a De Bruijn matrix. Moreover, we consider in particular the binary roots of Jn, i.e. the binary solutions to Am=Jn. On the one hand, we prov...

Vandanjav, Adiyasuren Undrakh, Batzorig
Published in
Linear Algebra and Its Applications

In this paper, we compute and compare the numerical radii of certain weighted shift matrices. Also we compute the numerical radius of a weighted shift operator on the Hardy space H2. The purpose of this paper is to develop results in [7].

Arav, Marina Hall, Frank J. Li, Zhongshan van der Holst, Hein Zhang, Lihua Zhou, Wenyan
Published in
Linear Algebra and Its Applications

A sign pattern matrix is a matrix whose entries are from the set {+,−,0}. If A is an m×n sign pattern matrix, the qualitative class of A, denoted Q(A), is the set of all real m×n matrices B=[bi,j] with bi,j positive (respectively, negative, zero) if ai,j is + (respectively, −, 0). The minimum rank of a sign pattern matrix A, denoted mr(A), is the m...

Mathai, A.M.
Published in
Linear Algebra and Its Applications

Fractional integral operators of the first and second kind in many and complex matrix-variate cases are the main topic of discussion here. After discussing fractional integral operators in the real scalar variable case, the ideas are extended to fractional integral operators involving one matrix variable in the real and complex cases. Then these id...

Alexandersson, Per Shapiro, Boris
Published in
Linear Algebra and Its Applications

Given an arbitrary complex-valued infinite matrix A=(aij), i=1,…,∞; j=1,…,∞ and a positive integer n we introduce a naturally associated polynomial basis BA of C[x0,…,xn]. We discuss some properties of the locus of common zeros of all polynomials in BA having a given degree m; the latter locus can be interpreted as the spectrum of the m×(m+n)-subma...

Garnett, C. Olesky, D.D. van den Driessche, P.
Published in
Linear Algebra and Its Applications

For a real n×n matrix A having n+ (n−) eigenvalues with positive (resp. negative) real part, nz zero eigenvalues and 2np nonzero pure imaginary eigenvalues, the refined inertia of A is ri(A)=(n+,n−,nz,2np). When n=3, let H3={(0,3,0,0),(0,1,0,2),(2,1,0,0)}. A 3×3 sign pattern A requires refined inertia H3 if {ri(A)|A has sign pattern A}=H3. Necessar...