Related papers: Reduction and Normal Forms of Matrix Pencils
We consider the numerical approximation of $f({\cal A})b$ where $b\in{\mathbb R}^{N}$ and $\cal A$ is the sum of Kronecker products, that is ${\cal A}=M_2 \otimes I + I \otimes M_1\in{\mathbb R}^{N\times N}$. Here $f$ is a regular function…
In this research, the Bernoulli polynomials are introduced. The properties of these polynomials are employed to construct the operational matrices of integration together with the derivative and product. These properties are then utilized…
A manifestly Lorentz-covariant calculus based on two matrix-coordinates and their associated derivatives is introduced. It allows formulating relativistic field theories in any even-dimensional spacetime. The construction extends a…
We study the Poincar\'e series of the mixed and pure trace rings of generic matrices. These series are known to be rational functions. We obtain an explicit formula in lowest terms in the case of $2\times2$ matrices; a denominator, which we…
We study sums of the form $\sum_{k=m}^n a_{nk} b_{km}$, where $a_{nk}$ and $b_{km}$ are binomial coefficients or unsigned Stirling numbers. In a few cases they can be written in closed form. Failing that, the sums still share many common…
A unified approach to the determination of eigenvalues and eigenvectors of specific matrices associated with directed graphs is presented. Matrices studied include the distance matrix, distance Laplacian, and distance signless Laplacian, in…
A Moore--Penrose inverse of an arbitrary complex matrix A is defined as a unique matrix A' such that AA'A=A, A'AA'=A', and AA', A'A are Hermite matrices. We show that this definition has a natural generalization in the context of shortly…
We define a general product of two $n$-dimensional tensors $\mathbb {A}$ and $\mathbb {B}$ with orders $m\ge 2$ and $k\ge 1$, respectively. This product is a generalization of the usual matrix product, and satisfies the associative law.…
Hermitian linear matrix pencils are ubiquitous in control theory, operator systems, semidefinite optimization, and real algebraic geometry. This survey reviews the fundamental features of the matricial solution set of a linear matrix…
Our focus is on the set of lower-triangular, infinite matrices that have natural operations like addition, multiplication by a number, and matrix multiplication. With respect to addition this set forms and abelian group while with respect…
A matrix pattern is often either a sign pattern with entries in {0,+,-} or, more simply, a nonzero pattern with entries in {0,*}. A matrix pattern A is spectrally arbitrary if for any choice of a real matrix spectrum, there is a real matrix…
In this article we study the combinatorics of congruence subgroups of the modular group. We consider the notion of minimal monomial solutions. These are the solutions of a matrix equation (also appearing in the study of Coxeter friezes),…
We present a new interpretation for reduced density matrices of secondary variables in relativistic systems via an analysis of Wigner's method to construct the irreducible unitary representations of the Poincar\'e group. We argue that the…
Matrix models of 2D quantum gravity are either exactly solvable for matter of central charge $ c\leq 1, $ or not understood. It would be useful to devise an approximate scheme which would be reasonable for the known cases and could be…
For cubic pencils we define the notion of an involution curve. This is a curve which intersects each curve of the pencil in exactly one non-base point of the pencil. Involution curves can be used to construct integrable maps of the plane…
We consider a pencil of matrix Sturm-Liouville operators on a finite interval. We study properties of its spectral characteristics and inverse problems that consist in recovering of the pencil by the spectral data: eigenvalues and…
We consider a quadratic matrix boundary value problem with equations and boundary conditions dependent on a spectral parameter. We study an inverse problem that consists in recovering the differential pencil by the so-called Weyl matrix. We…
Newton's inequalities $c_n^2 \ge c_{n-1}c_{n+1}$ are shown to hold for the normalized coefficients $c_n$ of the characteristic polynomial of any $M$- or inverse $M$-matrix. They are derived by establishing first an auxiliary set of…
We exhibit an explicit, deterministic algorithm for finding a canonical form for a positive definite matrix under unimodular integral transformations. We use characteristic sets of short vectors and partition-backtracking graph software.…
The main purpose of this paper is to develop new algorithms for computing invariant rings in a general setting. This includes invariants of nonreductive groups but also of groups acting on algebras over certain rings. In particular, we…