Loading...
Quaternionic conjugate-gradient method for solving the matrix equation AXB=C over generalized quaternions
Author(s)
Date Issued
October 1, 2026
Type
Article
Abstract
We propose a structure-exploiting conjugate-gradient (CG)–type algorithm for solving the matrix equation AXB=C over the generalized quaternions. The proposed method is developed from an idea of operating the linear map K(X)=AXB directly on the matrix space. This enables a matrix-free Krylov subspace implementation that avoids the explicit construction of the associated large-scale Kronecker matrix. By exploiting the intrinsic component-wise structure of quaternion matrices, the procedure performs all computations through matrix–matrix multiplications, leading to significant reductions in computational complexity and memory requirements. Finite-step convergence of the algorithm is established under suitable positive-definiteness assumptions. The proposed framework naturally includes real- and complex-valued matrix equations, the Hamilton quaternion case, and other quaternion algebras as special cases. Numerical experiments confirm the efficiency, robustness, and scalability of the proposed algorithm compared with existing methods.
Citation
Computational and Applied Mathematics, 45(9), 2026