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

The eigenvector-eigenvalue identity, formally named by Peter B. Denton, Stephen J. Parker, Terence Tao, Xining Zhang [Bull. Math. Amer. Soc., 2021], which is a basic and important identity in linear commutative algebra. 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.