A block GMRES method for the solution of the sylvester quaternion matrix equation

In this thesis work, we aim to obtain the solution of the Sylvester quaternion matrix equation AX-XB=C, where A∈H^(m×m), B∈H^(n×n), C∈H^(m×n) and m is very large such that m≫n. One difficulty in solving such an equation is that quaternion scalars are not commutative under multiplication. To overcome this difficulty, we exploit the complex matrix representations of quaternion matrices, and, consequently, turn the quaternion matrix equation into a complex matrix equation. By taking into account the fact that the resulting complex matrix equation involve large matrices, and m≫n, we tailor a block GMRES method, that makes use of certain Krylov subspaces. As the proposed block GMRES method operates on small Krylov subspaces, it requires the solution of small complex matrix equations. The Krylov subspaces are expanded and the small complex matrix equations are refined iteratively. In the end, our GMRES approach yields the complex representation of the solution of the Sylvester quaternion matrix equation at hand. In the final step of our approach, this complex matrix representation is converted into the corresponding quaternion matrix leading to the solution of the quaternion matrix equation AX-XB=C.