The eigenvector-eigenvalue identity for the quaternion matrix with its algorithm and computer program

TL;DR

This paper extends the eigenvector-eigenvalue identity to quaternion matrices, develops a new algorithm for eigenvector computation from eigenvalues, and provides a computer program demonstrating effective performance.

Contribution

It introduces a quaternion version of the eigenvector-eigenvalue identity, along with a novel algorithm and software implementation for eigenvector calculation.

Findings

The quaternion eigenvector-eigenvalue identity is successfully established.

The algorithm accurately computes eigenvectors from eigenvalues for quaternion Hermitian matrices.

The program demonstrates good performance on example cases.

Abstract

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 convenient and powerful tool to succinctly determine eigenvectors from eigenvalues. The identity relates the eigenvector component to the eigenvalues of $A$ and the minor $M_j$, which is formulated in an elegant form as \[ \lvert v_{i,j} \rvert^2\prod_{k=1;k\ne i}^{n-1}({\lambda_i}(A)-{\lambda_k}(A))=\prod_{k=1}^{n-1}({\lambda_i}(A)-{\lambda_k}(M_j)). \,\,\,%\mbox{(\cite{tao-eig,D-P-T-Z})} \] In fact, it has been widely applied in various fields such as numerical linear algebra, random matrix theory, inverse eigenvalue problem, graph theory, neutrino physics and so on. In this paper, we extend the eigenvector-eigenvalue identity to the quaternion division ring, which is non-commutative. A version of eigenvector-eigenvalue identity for the quaternion matrix is established. Furthermore, we give a new method and algorithm to compute the eigenvectors from the right eigenvalues for the quaternion Hermitian matrix. A program is designed to realize the algorithm to compute the eigenvectors. An open problem ends the paper. Some examples show a good performance of the algorithm and the program.