Related papers: Eigenbasis for a weighted adjacency matrix associa…

Two confluent rewriting systems in noncommutatives polynomials are constructed using the equations allowing the identification of the local coordinates (of second kind) of the graphs of the $\zeta$ polymorphism as being (shuffle or…

Matrix-valued analogues of the little q-Jacobi polynomials are introduced and studied. For the 2x2-matrix-valued little q-Jacobi polynomials explicit expressions for the orthogonality relations, Rodrigues formula, three-term recurrence…

The article presents weighted adjacency spectrum of complete multipartite graphs, characterize its families with three distinct eigenvalues and identifies integral matrices. Also, we observe that for almost all weighted matrices, the energy…

This is the second part of the project `unified theory of classical orthogonal polynomials of a discrete variable derived from the eigenvalue problems of hermitian matrices.' In a previous paper, orthogonal polynomials having Jackson…

The eight-vertex model on the square lattice with vertex weights $a,b,c,d$ obeying the relation $(a^2+ab)(b^2+ab)=(c^2+ab)(d^2+ab)$ is considered. Its transfer matrix with $L=2n+1,\, n\geqslant 0,$ vertical lines and periodic boundary…

We consider nonnormal matrix-valued dynamical systems with discrete time. For an eigenvalue of matrix, the number of times it appears as a root of the characteristic polynomial is called the algebraic multiplicity. On the other hand, the…

Evaluation of the eigenvectors of symmetric tridiagonal matrices is one of the most basic tasks in numerical linear algebra. It is a widely known fact that, in the case of well separated eigenvalues, the eigenvectors can be evaluated with…

We establish new explicit connections between classical (scalar) and matrix Gegenbauer polynomials, which result in new symmetries of the latter and further give access to several properties that have been out of reach before: generating…

We define the generalized basic hypergeometric polynomial of degree $N \geq 1$ in terms of the generalized basic hypergeometric function, which depends on (arbitrary, generic, possibly complex) parameters $q \neq 1$, the $r \geq 0$…

Let $M$ be a square matrix and let $p(t)$ be a monic polynomial of degree $n$. Let $Z$ be a set of $n\times n$ matrices. The multiplicative inverse eigenvalue problem asks for the construction of a matrix in $Z$ such that the product matrix…

Peter Denton, Stephen Parke, Terence Tao and Xining Zhang [arxiv 2019] presented a basic and important identity in linear commutative algebra, so-called {\bf the eigenvector-eigenvalue identity} (formally named in [BAMS, 2021]), which is a…

Given a graph $G$, we have the adjacency matrix $A(G)$ and degree diagonal matrix $D(G)$. The $Q$-spectrum is the all eigenvalues of $Q$-matrix $Q(G)=A(G)+D(G)$. A class of graphs is determined by their generalized $Q$-spectrum (DGQS for…

To measure an observable of a quantum mechanical system leaves it in one of its eigenstates and the result of the measurement is one of its eigenvalues. This process is shown to be a computational resource. It allows one, in principle, to…

We introduce the notion of "hypergeometric" polynomials with respect to Newtonian bases. These polynomials are eigenfunctions ($L P_n(x) = \lambda_n P_n(x)$) of some abstract operator $L$ which is 2-diagonal in the Newtonian basis…

We explore algebraic and spectral properties of weighted graphs containing twin vertices that are useful in quantum state transfer. We extend the notion of adjacency strong cospectrality to arbitrary Hermitian matrices, with focus on the…

We describe bivariate polynomial sequences orthogonal to a symmetric weight function in terms of several bivariate polynomial sequences orthogonal with respect to Christoffel transformations of the initial weight under a quadratic…

The paper is devoted to the generalization of Lusztig's q-analog of weight multiplicities to the Lie superalgebras gl(n,m) and spo(2n,M). We define such q-analogs K_{lambda,mu}(q) for the typical modules and for the irreducible covariant…

Let $\mathbb{F}_q$ denote a finite field with $q$ elements. Let $N$ and $D$ denote integers with $N>D \ge 1$. Let $\mathcal{V}$ denote an $N$-dimensional vector space over $\mathbb{F}_q$. The Grassmann graph $J_q(N,D)$ is the graph with…

The current general form of the well-known Eigenvalue Interlacing Theorem states that, given an $N \times N$ Hermitian matrix $P$, the eigenvalues of the matrix product $Q^{H} P Q$ will interlace those of $P$ if the columns of the $N \times…

Let $\Gamma$ denote a distance-regular graph with diameter $D \ge 3$. Assume $\Gamma$ has classical parameters $(D,b,\alpha,\beta)$ with $b < -1$. Let $X$ denote the vertex set of $\Gamma$ and let $A \in MX$ denote the adjacency matrix of…