Solution of a Sylvester Quaternion Matrix Equation by a Block Krylov Subspace Method

We focus on the solution of the Sylvester quaternion matrix equation$\ AX-XB=C$, where $A\in {\mathbb{H}}^{m\times m}$, $B\in {\mathbb{H}}^{n\times n}$, $C\in {\mathbb{H}}^{m\times n}$ and $m$ is very large such that $m\gg n$. Non-commutative nature of quaternion scalars under multiplication is a hurdle in the solution of such a matrix equation. Thus, instead of directly dealing with the quaternion matrix equation, we make use of the complex matrix representations of quaternion matrices, and turn the quaternion matrix equation into a complex matrix equation of size twice as big. Since the resulting complex matrix equation involves large matrices, assuming $m$ is large, in particular$\ m\gg n$, we present a block Generalized Minimal Residual (GMRES) method that seeks the solution of the complex matrix equation in small affine spaces defined in terms of Krylov subspaces. The solution in such a small affine space can equivalently be posed as the solution of a small complex matrix equation, which can be solved directly for instance by rewriting it as a linear system. At every iteration of our block GMRES method, the Krylov subspaces are expanded with the addition of new vectors, and the small complex matrix equations are altered accordingly. Our block GMRES method eventually produces the complex representation of an approximate solution of the original Sylvester quaternion matrix equation. Finally, this complex matrix representation is transformed back into the corresponding quaternion matrix, which is an approximate solution of the original quaternion matrix equation $AX-XB=C$.