A Global Krylov Subspace Method for the Sylvester Quaternion Matrix Equation

Abstract

This study concerns the Sylvester matrix equation in the quaternion setting when the coefficient matrices as well as the unknown matrix have quaternion entries. We propose a global Generalized Minimal Residual (GMRES) method for the solution of such a matrix equation. The proposed approach works directly with the Sylvester operator to generate orthonormal bases for Krylov subspaces formed of matrices. Then, the best approximate matrix solution to the Sylvester equation at hand in such a Krylov subspace is constructed from a matrix minimizing the Frobenius norm of the residual. We describe how this minimization of the residual norm can be carried out efficiently and report numerical results on real examples related to image restoration.

Keywords

• Sylvester quaternion matrix equation
• quaternion Krylov subspace
• global GMRES
• quaternion Arnoldi process

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