Exact and least-squares solutions of a generalized Sylvester-transpose matrix equation over generalized quaternions - Janthip Jaiprasert; Pattrawut Chansangiam (My University, 2024)
We have considered a generalized Sylvester-transpose matrix equation $ AXB + CX^TD = E, $ where $ A, B, C, D, $ and $ E $ are given rectangular matrices over a generalized quaternion skew-field, and $ X $ is an unknown matrix. We have applied certain vectorizations and real representations to transform the matrix equation into a matrix equation over the real numbers. Thus, we have investigated a solvability condition, general exact/least-squares solutions, minimal-norm solutions, and the exact/least-squares solution closest to a given matrix. The main equation included the equation $ AXB = E $ and the Sylvester-transpose equation. Our results also covered such matrix equations over the quaternions, and quaternionic linear systems.
Keywords: Transpose; Sylvester equation; Matrix (chemical analysis); Sylvester matrix; MoorePenrose pseudoinverse; Least-squares function approximation; Matrix Theory and Algorithms