Fast algorithms for eigenvalues of quaternion matrices and their applications

This report briefly analyzes the wide range of applications of quaternion matrix computation in modern science and engineering, including problems in color image processing, three-dimensional signal processing, spatial rotation and quaternion quantum mechanics. The quaternion matrix eigenvalue problem is not only an important branch of numerical linear algebra, but also a fundamental problem in fields such as applied mathematics and physics. However, the computational efficiency of the existing algorithms is relatively low, which limits their large-scale popularization in practical applications.
This report develops three more efficient algorithms for computing the eigenvalues of quaternion matrices based on the algebraic structure and properties of quaternion matrices. Numerical experiments show that the proposed algorithms have high computational efficiency and are successfully applied to optimize an existing color face recognition scheme based on quaternion matrix eigenvalues. Simulation experiments further validate the significant performance improvement of the algorithms in practical applications.
In summary, this report not only optimizes the performance of the algorithm in the field of quaternion numerical computation, but also provides theoretical support and foundation for applications based on quaternion matrix eigenvalues, as well as other potential technical fields.
This work is supported by the Ministry of Science and Higher Education of the Russian Federation, supplementary agreement (No. 075-02-2025-1792, March 11, 2025), and by the Russian Science Foundation grant No. 23-71-30013 (https://rscf.ru/project/23-71-30013/).