Related papers: On the Right Eigenvalues of the Quaternionic Matri…
Recovering hidden structures from incomplete or noisy data remains a pervasive challenge across many fields, particularly where multi-dimensional data representation is essential. Quaternion matrices, with their ability to naturally model…
Criterion for a companion matrix to have a certain number of flat portions on the boundary of its numerical range is given. The criterion is specialized to the cases of 3-by-3 and 4-by-4 matrices. In the latter case, it is proved that a…
Due to the non-commutative nature of quaternions we introduce the concept of left and right action for quaternionic numbers. This gives the opportunity to manipulate appropriately the $H$-field. The standard problems arising in the…
We find in our quaternionic version of the electroweak theory an apparently hopeless problem: In going from complex to quaternions, the calculation of the real-valued parameters of the CKM matrix drastically changes. We aim to explain this…
Motivated by a quaternionic formulation of quantum mechanics, we discuss quaternionic and complex linear differential equations. We touch only a few aspects of the mathematical theory, namely the resolution of the second order differential…
In this short note, we present a novel method for computing exact lower and upper bounds of eigenvalues of a symmetric tridiagonal interval matrix. Compared to the known methods, our approach is fast, simple to present and to implement, and…
We study the average condition number for polynomial eigenvalues of collections of matrices drawn from various random matrix ensembles. In particular, we prove that polynomial eigenvalue problems defined by matrices with Gaussian entries…
We present an explicit basis for orders of arbitrary level N>1 in definite rational quaternion algebras. These orders have applications to computations of spaces of elliptic and quaternionic modular forms.
In this article we give bounds for the eigenvalues of a matrix, which can be seen as a common generalization of meet and join matrices and therefore also as a generalization of both GCD and LCM matrices. Although there are some results…
The complex matrix representation for a quaternion matrix is used in this paper to find necessary and sufficient conditions for the existence of an $H$-selfadjoint $m$th root of a given $H$-selfadjoint quaternion matrix. In the process,…
We give an elementary introduction to some recent polyhedral techniques for understanding and solving systems of multivariate polynomial equations. We provide numerous concrete examples and illustrations, and assume no background in…
Let $D$ be an integrally closed domain with quotient field $K$ and $n$ a positive integer. We give a characterization of the polynomials in $K[X]$ which are integer-valued over the set of matrices $M_n(D)$ in terms of their divided…
We obtain several norm and eigenvalue inequalities for positive matrices partitioned into four blocks. The results involve the numerical range of the off-diagonal block X, especially the distance from 0 to W(X).
In this work we present a method, based on the use of Bernstein polynomials, for the numerical resolution of some boundary values problems. The computations have not need of particular approximations of derivatives, such as finite…
For signed graphs we provide a cubic polynomial upper bound on the multiplicity of its eigenvalues. We show that this bound is sharp by providing examples of signed graphs in which it is attained. We also discuss particular cases in which…
We show that various old and new bounds involving eigenvalues of a complex n x n matrix are immediate consequences of the inequalities involving variance of real and complex numbers.
We lay down the foundations of the Eigenvalue Method in coding theory. The method uses modern algebraic graph theory to derive upper bounds on the size of error-correcting codes for various metrics, addressing major open questions in the…
We provide upper bounds for the cardinality of the value set of a polynomial map in several variables over a finite field. These bounds generalize earlier bounds for univariate polynomials.
Consider an $n\times n$ matrix polynomial $P(\lambda)$ and a set $\Sigma$ consisting of $k \le n$ distinct complex numbers. In this paper, a (weighted) spectral norm distance from $P(\lambda)$ to the matrix polynomials whose spectra include…
We provide a link between the virial theorem in functional analysis and the method of multipliers in theory of partial differential equations. After giving a physical insight into the techniques, we show how to use them to deduce the…