Exact and least-squares solutions of a generalized Sylvester-transpose matrix equation over generalized quaternions

We have considered a generalized Sylvester -transpose matrix equation AXB + CXTD = 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.