2022/02/16 by Yuchao He, He, Yuchao, Mengda Wu +3
Computer Science · Mathematics · #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Graph theory and applications #Matrix Theory and Algorithms #Numerical Analysis (math.NA) #Rings and Algebras (math.RA) #Tensor decomposition and applications
paper · pdf · doi:10.48550/arxiv.2203.02273
openalex publication_date 2022/02/16 · openalex created_date 2022/04/26 · openalex updated_date 2026/07/28
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 Mj, which is formulated in an elegant form as | vi,j |2∏k=1;k≠ in-1(λi(A)-λk(A))=∏k=1n-1(λi(A)-λk(Mj)). %\mbox(\citetao-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.