Related papers: Right eigenvalue equation in quaternionic quantum …
We discuss the (right) eigenvalue equation for $\mathbb{H}$, $\mathbb{C}$ and $\mathbb{R}$ linear quaternionic operators. The possibility to introduce an isomorphism between these operators and real/complex matrices allows to translate the…
This paper presents an innovative set of tools developed to support a methodology to find the left eigenvalues of $m$ order quaternion square matrix. It is solving four real polynomial equations of order not greater than $4m-3$ in four…
The poses of $m$ robotics in $n$ time points may be represented by an $m \times n$ dual quaternion matrix. In this paper, we study the spectral theory of dual quaternion matrices. We introduce right and left eigenvalues for square dual…
By using a real matrix translation, we propose a coupled eigenvalue problem for octonionic operators. In view of possible applications in quantum mechanics, we also discuss the hermiticity of such operators. Previous difficulties in…
We present a practical Newton-based method for computing left eigenvalues of quaternion matrices. It uses only standard real/complex linear-algebra kernels via embeddings and applies to matrices of any size. Extensive tests on literature…
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…
This paper deals with eigenvalues and eigenvectors of bicomplex linear operators defined on bicomplex space. We investigate the properties of these operators in the context of eigenvalues and eigenvectors, along with some relevant theorems.…
In a recent paper [J.Math.Phys. vol42, 2236-2265 (2001)], we discussed differential operators within a quaternionic formulation of quantum mechanics. In particular, we proposed a practical method to solve quaternionic and complex linear…
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…
It is known that a $2\times 2$ quaternionic matrix has one, two or an infinite number of left eigenvalues, but the available algebraic proofs are difficult to generalize to higher orders. In this paper a different point of view is adopted…
We introduce right eigenvalues and subeigenvalues for square dual complex matrices. An $n \times n$ dual complex Hermitian matrix has exactly $n$ right eigenvalues and subeigenvalues, which are all real. The Hermitian matrix is positive…
The renewed interest in investigating quaternionic quantum mechanics, in particular tunneling effects, and the recent results on quaternionic differential operators motivate the study of resolution methods for quaternionic differential…
We study the Rayleigh quotient of a Hermitian matrix with quaternionic coefficients and prove its main properties. As an application, we give some relationships between left and right eigenvalues of Hermitian and symplectic matrices.
We obtain a complete characterization of the $2\times 2$ symplectic matrices having an infinite number of left eigenvalues. Previously, we give a new proof of a result from Huang and So about the number of eigenvalues of a quaternionic…
A class of non-linear eigenvalue problems defined in the form of operator polynomials is investigated. The problems are related to wave equations which appear in a relativistic quantum field theory. Spectral asymptotics for this class are…
In order to obtain a consistent formulation of octonionic quantum mechanics (OQM), we introduce left-right barred operators. Such operators enable us to find the translation rules between octonionic numbers and $8\times 8$ real matrices (a…
In order to obtain a consistent formulation of octonionic quantum mechanics (OQM), we introduce left-right barred operators. Such operators enable us to find the translation rules between octonionic numbers and $8\times 8$ real matrices (a…
In (relativistic) electronic structure methods, the quaternion matrix eigenvalue problem and the linear response (Bethe-Salpeter) eigenvalue problem for excitation energies are two frequently encountered structured eigenvalue problems.…
This paper is devoted to the study of the S-eigenvalue of finite type of a bounded right quaternionic linear operator acting in a right quaternionic Hilbert space. The study is based on the different properties of the Riesz projection…
We describe an algorithm to compute the extremal eigenvalues and corresponding eigenvectors of a symmetric matrix by solving a sequence of Quadratic Binary Optimization problems. This algorithm is robust across many different classes of…