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Calculation of the greatest common divisor of perturbed polynomials

Zítko, Jan, Eliaš, Ján (2013)

Programs and Algorithms of Numerical Mathematics

The coefficients of the greatest common divisor of two polynomials $f$ and $g$ (GCD $(f, g)$ ) can be obtained from the Sylvester subresultant matrix $S_{j} (f, g)$ transformed to lower triangular form, where $1 \leq j \leq d$ and $d =$ deg(GCD $(f, g)$ ) needs to be computed. Firstly, it is supposed that the coefficients of polynomials are given exactly. Transformations of $S_{j} (f, g)$ for an arbitrary allowable $j$ are in details described and an algorithm for the calculation of the GCD $(f, g)$ is formulated. If inexact polynomials are given, then an approximate greatest...

Cayley-Hamilton theorem for left eigenvalues of 3 × 3 quaternionic matrices

E. Macías-Virgós, M.J. Pereira-Sáez (2014)

Special Matrices

We prove that any quaternionic matrix of order n ≤3 admits a characteristic function, whose roots are the left eigenvalues, that satisfes Cayley-Hamilton theorem.

Central limit theorems for eigenvalues of deformations of Wigner matrices

M. Capitaine, C. Donati-Martin, D. Féral (2012)

Annales de l'I.H.P. Probabilités et statistiques

In this paper, we study the fluctuations of the extreme eigenvalues of a spiked finite rank deformation of a Hermitian (resp. symmetric) Wigner matrix when these eigenvalues separate from the bulk. We exhibit quite general situations that will give rise to universality or non-universality of the fluctuations, according to the delocalization or localization of the eigenvectors of the perturbation. Dealing with the particular case of a spike with multiplicity one, we also establish a necessary and...

Changes in signature induced by the Lyapunov mapping $𝔏_{A} : X \to A X + X A^{*}$ .

Haynes, Tyler (1989)

International Journal of Mathematics and Mathematical Sciences

Characterization of α1 and α2-matrices

Rafael Bru, Ljiljana Cvetković, Vladimir Kostić, Francisco Pedroche (2010)

Open Mathematics

This paper deals with some properties of α1-matrices and α2-matrices which are subclasses of nonsingular H-matrices. In particular, new characterizations of these two subclasses are given, and then used for proving algebraic properties related to subdirect sums and Hadamard products.

Characterizing matrices with $𝐗$ -simple image eigenspace in max-min semiring

Ján Plavka, Sergeĭ Sergeev (2016)

Kybernetika

A matrix $A$ is said to have $𝐗$ -simple image eigenspace if any eigenvector $x$ belonging to the interval $𝐗 = {x : \underset{̲}{x} \leq x \leq \bar{x}}$ is the unique solution of the system $A \otimes y = x$ in $𝐗$ . The main result of this paper is a combinatorial characterization of such matrices in the linear algebra over max-min (fuzzy) semiring. The characterized property is related to and motivated by the general development of tropical linear algebra and interval analysis, as well as the notions of simple image set and weak robustness (or weak stability) that...

Circulant matrices.

Lara, Teodoro (2001)

Divulgaciones Matemáticas

Coalescing Fiedler and core vertices

Didar A. Ali, John Baptist Gauci, Irene Sciriha, Khidir R. Sharaf (2016)

Czechoslovak Mathematical Journal

The nullity of a graph $G$ is the multiplicity of zero as an eigenvalue in the spectrum of its adjacency matrix. From the interlacing theorem, derived from Cauchy’s inequalities for matrices, a vertex of a graph can be a core vertex if, on deleting the vertex, the nullity decreases, or a Fiedler vertex, otherwise. We adopt a graph theoretical approach to determine conditions required for the identification of a pair of prescribed types of root vertices of two graphs to form a cut-vertex of unique...

Coefficients of ergodicity generated by non-symmetrical vector norms

Antonín Lešanovský (1990)

Czechoslovak Mathematical Journal

Completing block Hermitian matrices with maximal and minimal ranks and inertias.

Tian, Yongge (2010)

ELA. The Electronic Journal of Linear Algebra [electronic only]

Computing codimensions and generic canonical forms for generalized matrix products.

Kagstrom, Bo, Karlsson, Lars, Kressner, Daniel (2011)

ELA. The Electronic Journal of Linear Algebra [electronic only]

Computing the CS and the Generalized Singular Value Decompositions.

Charles Van Loan (1985)

Numerische Mathematik

Concentration of the spectral measure for large matrices.

Guionnet, A., Zeitouni, O. (2000)

Electronic Communications in Probability [electronic only]

Conjunctively polynomial-like Boolean functions.

Gonda, J. (2007)

Acta Mathematica Academiae Paedagogicae Nyí regyháziensis. New Series [electronic only]

Connectivity, toughness, spanning trees of bounded degree, and the spectrum of regular graphs

Sebastian M. Cioabă, Xiaofeng Gu (2016)

Czechoslovak Mathematical Journal

The eigenvalues of graphs are related to many of its combinatorial properties. In his fundamental work, Fiedler showed the close connections between the Laplacian eigenvalues and eigenvectors of a graph and its vertex-connectivity and edge-connectivity. We present some new results describing the connections between the spectrum of a regular graph and other combinatorial parameters such as its generalized connectivity, toughness, and the existence of spanning trees with bounded degree.

Constructing copositive matrices from interior matrices.

Johnson, Charles R., Reams, Robert (2008)

ELA. The Electronic Journal of Linear Algebra [electronic only]

Constructions of trace zero symmetric stochastic matrices for the inverse eigenvalue problem.

Reams, Robert (2002)

ELA. The Electronic Journal of Linear Algebra [electronic only]

Controllability of linear impulsive systems – an eigenvalue approach

Vijayakumar S. Muni, Raju K. George (2020)

Kybernetika

This article considers a class of finite-dimensional linear impulsive time-varying systems for which various sufficient and necessary algebraic criteria for complete controllability, including matrix rank conditions are established. The obtained controllability results are further synthesised for the time-invariant case, and under some special conditions on the system parameters, we obtain a Popov-Belevitch-Hautus (PBH)-type rank condition which employs eigenvalues of the system matrix for the investigation...

Controllability of partially prescribed matrices.

Gloria Cravo (2009)

Collectanea Mathematica

Controllable and tolerable generalized eigenvectors of interval max-plus matrices

Matej Gazda, Ján Plavka (2021)

Kybernetika

By max-plus algebra we mean the set of reals $ℝ$ equipped with the operations $a \oplus b = max {a, b}$ and $a \otimes b = a + b$ for $a, b \in ℝ .$ A vector $x$ is said to be a generalized eigenvector of max-plus matrices $A, B \in ℝ (m, n)$ if $A \otimes x = λ \otimes B \otimes x$ for some $λ \in ℝ$ . The investigation of properties of generalized eigenvectors is important for the applications. The values of vector or matrix inputs in practice are usually not exact numbers and they can be rather considered as values in some intervals. In this paper the properties of matrices and vectors with inexact (interval) entries...

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