A Fast Projected Gradient Algorithm for Quaternion Hermitian Eigenvalue Problems

Shan-Qi Duan and Qing-Wen Wang ()
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Shan-Qi Duan: Department of Mathematics and Newtouch Center for Mathematics, Shanghai University, Shanghai 200444, China
Qing-Wen Wang: Department of Mathematics and Newtouch Center for Mathematics, Shanghai University, Shanghai 200444, China

Mathematics, 2025, vol. 13, issue 6, 1-14

Abstract: In this paper, based on the novel generalized Hamilton-real (GHR) calculus, we propose for the first time a quaternion Nesterov accelerated projected gradient algorithm for computing the dominant eigenvalue and eigenvector of quaternion Hermitian matrices. By introducing momentum terms and look-ahead updates, the algorithm achieves a faster convergence rate. We theoretically prove the convergence of the quaternion Nesterov accelerated projected gradient algorithm. Numerical experiments show that the proposed method outperforms the quaternion projected gradient ascent method and the traditional algebraic methods in terms of computational accuracy and runtime efficiency.

Keywords: GHR calculus; principal eigenvalues; quaternion Hermitian matrix; Nesterov’s accelerated gradient method (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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