相关论文: Finding Octonionic Eigenvectors Using Mathematica

This work rests upon the certainty that only fields of real and complex numbers, quaternions and octonions have algebras of all four arithmetical operations. Also quaternions are good to represent 3-dimensional Euclid space and…

Working over the split octonions over an algebraically closed field, we solve all polynomial equations in which all the coefficients but the constant term are scalar. As a consequence, we calculate the n-th roots of an octonion.

In this paper we express the eigenvalues of a sort of real heptadiagonal symmetric matrices as the zeros of explicit rational functions establishing upper and lower bounds for each of them. From these prescribed eigenvalues we compute also…

In this paper, we consider the operator properties of various phononic eigenvalue problems. We aim to answer some fundamental questions about the eigenvalues and eigenvectors of phononic operators. These include questions about the…

The Hermitian eigenvalue problem asks for the possible eigenvalues of a sum of $n\times n$ Hermitian matrices, given the eigenvalues of the summands. The regular faces of the cones $\Gamma_n(s)$ controlling this problem have been…

It is shown that Pythagorean triples can be used to generate matrices that have integer eigenvalues for all permutations of their coefficients, via simple formulas. For example, each and every permutation of the $2\times2$ matrix…

In this paper, we present some applications of quaternions and octonions. We present the real matrix representations for complex octonions and some of their properties which can be used in computations where these elements are involved.…

In recent years, there is a growing interest in the studying octonions, which are 8-dimensional hypercomplex numbers forming the biggest normed division algebras over the real numbers. In particular, various tools of the classical complex…

We study the real algebraic variety of real symmetric matrices with eigenvalue multiplicities determined by a partition. We present formulas for the dimension and Euclidean distance degree. We give a parametrization by rational functions.…

I investigate on the number t of real eigenvectors of a real symmetric tensor. In particular, given a homogeneous polynomial f of degree d in 3 variables, i prove that t is greater or equal than 2c+1, if d is odd and t is greater or equal…

We present a five-step method for the calculation of eigenvalue correlation functions for various ensembles of real random matrices, based upon the method of (skew-) orthogonal polynomials. This scheme systematises existing methods and also…

Intertwining analysis, algebra, numerical analysis and optimization, computing conjugate co-gradients of real-valued quotients gives rise to eigenvalue problems. In the linear Hermitian case, by inspecting optimal quotients in terms of…

Solving linear systems and computing eigenvalues are two fundamental problems in linear algebra. For solving linear systems, many efficient quantum algorithms have been discovered. For computing eigenvalues, currently, we have efficient…

In many applications, it is desired to obtain extreme eigenvalues and eigenvectors of large Hermitian matrices by efficient and compact algorithms. In particular, orthogonalization-free methods are preferred for large-scale problems for…

In this paper, we are interested in matrix valued orthogonal polynomials on the real line with respect to exponential weights. We obtain strong asymptotics as the degree tends to infinity in different regions of the complex plane, as well…

A hermitian matrix can be parametrized by a set consisting of its determinant and the eigenvalues of its submatrices. We established a group of equations which connect these variables with the mixing parameters of diagonalization. These…

The product of a Hermitian matrix and a positive semidefinite matrix has only real eigenvalues. We present bounds for sums of eigenvalues of such a product.

A remarkable mathematical property -- somehow hidden and recently rediscovered -- allows obtaining the eigenvectors of a Hermitian matrix directly from their eigenvalues. That opens the possibility to get the wavefunctions from the…

We present a fast Jacobi-like algorithm for computing the eigenvalues, and optionally the eigenvectors, of a real normal matrix. The method gains a computational advantage by using Paardekooper's method for skew-symmetric matrices The…

This paper describes a set of rational filtering algorithms to compute a few eigenvalues (and associated eigenvectors) of non-Hermitian matrix pencils. Our interest lies in computing eigenvalues located inside a given disk, and the proposed…