On the power method for quaternion right eigenvalue problem

In this paper, we study the power method of the right eigenvalue problem of a quaternion matrix A. If A is Hermitian, we propose the power method that is a direct generalization of that of complex Hermitian matrix. When A is non-Hermitian, by applying the properties of quaternion right eigenvalues, we propose the power method for computing the standard right eigenvalue with the maximum norm and the associated eigenvector. We also briefly discuss the inverse power method and shift inverse power method for the both cases. The real structure-preserving algorithm of the power method in the two cases are also proposed, and numerical examples are provided to illustrate the efficiency of the proposed power method and inverse power method.