Peter Denton, Stephen Parker, 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.

翻译：Peter Denton, Stephen Denton, Stephen Parker, Terence Tao 和 Xining Zhang [arxiv 2019] 展示了线性通象代数中的基本和重要身份,即所谓的igenvector-eigenvalid idate}(在[BAMS, 2021] 中正式命名),这是一个方便和有力的工具,可以简明地确定来自egenvalue的源数。 身份将igenctor 组件与$A 和小M_j$的元值联系起来, 以优雅的形式制作成[\ lverver v ⁇ i,j}\rverver2\\ prod ⁇ k=1;k\ne igencent- engencial distruals 程序。