An Iterative Algorithm for Quaternion Eigenvalue Problems in Signal Processing

This letter proposes a quaternion projection gradient ascent (QPGA) iterative algorithm based on generalized $\mathbb {HR}$ calculus for computing the principal eigenvalues and its eigenvectors of quaternion Hermitian matrices. We also prove the convergence of the QPGA algorithm, demonstrating that the estimated sequence of principal eigenvalues is monotonically increasing. Numerical experiments demonstrate the superiority of the proposed iterative method over traditional algebraic methods in terms of accuracy and speed, as well as the application of principal eigenvalues and their eigenvectors obtained by the QPGA algorithm in denoising with quaternion principal component analysis and quaternion least mean square (QLMS) algorithms in filtering fetal electrocardiograms. Overall, the fast quaternion eigenvalue solving method provides a novel and effective technical tool for quaternion signal processing.