Related papers: Finding Octonionic Eigenvectors Using Mathematica
We introduce variational methods for finding approximate eigenfunctions and eigenvalues of quantum Hamiltonians by constructing a set of orthogonal wave functions which approximately solve the eigenvalue equation.
We represent low dimensional quantum mechanical Hamiltonians by moderately sized finite matrices that reproduce the lowest O(10) boundstate energies and wave functions to machine precision. The method extends also to Hamiltonians that are…
We consider the problem of recovering a unitary eigendecomposition of a complex unitary matrix from that of its embedded real-valued formulation. Such formulations arise naturally in scientific computing workflows that employ…
Solving large-scale eigenvalue problems poses a significant challenge due to the computational complexity and limitations on the parallel scalability of the orthogonalization operation, when many eigenpairs are required. In this paper, we…
The affine inverse eigenvalue problem consists of identifying a real symmetric matrix with a prescribed set of eigenvalues in an affine space. Due to its ubiquity in applications, various instances of the problem have been widely studied in…
Dual quaternion matrices have various applications in robotic research and its spectral theory has been extensively studied in recent years. In this paper, we extend Jacobi method to compute all eigenpairs of dual quaternion Hermitian…
We show that a non-Hermitian operator with a tridiagonal matrix representation in a finite-dimensional vector space is similar to an Hermitian operator. The required condition is sufficient and simple examples show that it is not necessary.…
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…
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…
A simple iterative scheme is proposed for locating the parameter values for which a 2-parameter family of real symmetric matrices has a double eigenvalue. The convergence is proved to be quadratic. An extension of the scheme to complex…
For standard eigenvalue problems, a closed-form expression for the condition numbers of a multiple eigenvalue is known. In particular, they are uniformly 1 in the Hermitian case, and generally take different values in the non-Hermitian…
Geometrically, the eigenvectors of a square matrix $\mathbf{A}$ are not rotated by $\mathbf{A}$. Here we consider vectors that are rotated $\pi/2$ by $\mathbf{A}$; that is, vectors orthogonal to their images. We call these vectors…
We give formulae for first and second derivatives of generalized eigenvalues/eigenvectors of symmetric matrices and generalized singular values/singular vectors of rectangular matrices when the matrices are linear or nonlinear functions of…
Univariate polynomial root-finding is a classical subject, still important for modern computing. Frequently one seeks just the real roots of a polynomial with real coefficients. They can be approximated at a low computational cost if the…
We present an iteration for the computation of simple eigenvalues using a pseudospectrum approach. The most appealing characteristic of the proposed iteration is that it reduces the computation of a single eigenvalue to a small number of…
Estimating the number of eigenvalues located in a given interval of a large sparse Hermitian matrix is an important problem in certain applications and it is a prerequisite of eigensolvers based on a divide-and-conquer paradigm. Often an…
Many fields of science and engineering require finding eigenvalues and eigenvectors of large matrices. The solutions can represent oscillatory modes of a bridge, a violin, the disposition of electrons around an atom or molecule, the…
For any orthogonal polynomials system on real line we construct an appropriate oscillator algebra such that the polynomials make up the eigenfunctions system of the oscillator hamiltonian. The general scheme is divided into two types: a…
This paper is concerned with two extremal problems from matrix analysis. One is about approximating the top eigenspaces of a Hermitian matrix and the other one about approximating the orthonormal polar factor of a general matrix. Tight…
Differential equations with constant and variable coefficients over octonions are investigated. It is found that different types of differential equations over octonions can be resolved. For this purpose non-commutative line integration is…