In relation with an observation problem I have the matrix equation
(1) $XA + BX + CXD = E$
where all the matrices $A$, $B$, $C$, $D$, $E$ can be assumed real, square and known, whereas $X$ is the unknown matrix to be solved for. As discussed here: https://www.researchgate.net/post/How_do_I_solve_a_sylvester-unlike_Linear_Matrix_Equation_AX_XB_CXDE, and also in [1], this equation can in practice(if a solution exists) be solved(albeit possibly inefficiently) by applying the Kronecker tensor product to construct a standard linear system of equations defined as
(2) $\{A\otimes I + I\otimes B^T + C\otimes D^T\}v(X) = v(E)$
where $v(X)$, $v(E)$ denote the vectorizations of $X$, $E$, respectively. It was shown in [2] that the simpler Sylvester equation
(3) $XA$ + $BX$ = $C$
is universally solvable if $-A$ and $B$ have no common eigenvalues, while the more general equation
(4) $AXB + CXD = E$
is shown in f.ex. [1] to be solvable given that
(5a) $(A + \lambda C), (D - \lambda B)$ are regular pencils.
(5b)$\sigma(A,-C) \cap \sigma(D,B) = \emptyset$
where $\sigma(A,-C)$ and $\sigma(D,B)$ denote the spectra of the pencils in (5a), respectively. My question is, what would the conditions be for the solution of (1) to exist?
Any help would be appreciated!
[1] Chu, King-wah Eric. "The solution of the matrix equations AXB− CXD= E AND (YA− DZ, YC− BZ)=(E, F)." Linear Algebra and its Applications 93 (1987): 93-105.
[2] J.J. Sylvester. Sur la solution du cas le plus général des équations linéaires en quantités binaires, c’est-à-dire en quaternions ou en matrices du second ordres. Sur la résolution générale de l’équation linéaire en matrices d’un ordre quelconque. Sur l’équation linéaire trinôme en matrices d’un ordre quelconque. Comptes Rendus de l’Académie des Sciences, 99:117–118, 409–412, 432–436, 527–529, 1884.
