Solution to the generalised Sylvester matrix equation AV+BW=EVF

On solutions of matrix equation XF−AX=C and $XF-A\widetilde{X}=C$ over quaternion field

“…Definition 1 (Sylvester sum [23]). Let T (s) = t i=0 T i s i ∈ C m×q [s], F ∈ C p×p , and Z ∈ C q×p .…”

Solving the generalized Sylvester matrix equation AV + BW = V F via Kronecker map

“…Based on the above property of the Kronecker map, the following conclusions can be obtained. These conclusions can be found in [16]. …”

“…The following lemma gives the important property of the Kronecker map, which can be found in [16]. Based on the above property of the Kronecker map, the following conclusions can be obtained.…”

“…The concept of the Kronecker map was first introduced in [16] based on the underlying idea of [17]. In [16], the Kronecker map is utilized to prove the completeness of the proposed solution. Different from the idea in [16], in this note the Kronecker map is directly applied to obtaining the solution to the generalized Sylvester matrix equation (1).…”

Solving the generalized Sylvester matrix equation AV+BW=EVF via a Kronecker map

“…Besides, when the matrix F is an arbitrary matrix, a neat parametric solution is presented in [15] for the matrix equation (1) in terms of an R-controllability matrix, an observability matrix and a so-called generalized symmetric operator matrix. This solution has been used in [14] to design a kind of observer for descriptor linear systems.…”