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

We have considered a generalized Sylvester-transpose matrix equation $AXB+C{X}^{T}D=E,$ 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.