Least-Squares Solutions of Generalized Sylvester-Type Quaternion Matrix Equations

This paper focuses on finding solutions of generalized Sylvester type matrix equations over the quaternion skew-field. We express the general least-squares solutions, and perhermitian, skew-perhermitian least-squares solutions of AXB+CY D = E and AXB+CXD = E over the quaternion skew-field in terms of a vec operator (defined specifically for matrices over the quaternion skew-field) and the Moore-Penrose pseudoinverse. In addition, characterizations that facilitate the computation of the least-squares solutions closest to prescribed quaternion matrices are deduced. We illustrate our theoretical findings on several numerical examples, most of which originate from color image restoration via Tikhonov regularization.