A Sylvester equation is a matrix equation of the form
 |
where 
is 
, 
is 
, 
is 
, and the 
matrix 
is to be determined. The matrices
may have real or complex
entries. The equation has a unique solution for every 
iff no eigenvalue
of 
is the negative of an eigenvalue of 
, or equivalently, if the matrix
spectra of 
and 
are disjoint.
Using matrix vectorization to stack the columns of 
gives the equivalent linear system of equations
![[I_n tensor A+B^T tensor I_m]vec(X)=vec(C),](/images/equations/SylvesterEquation/NumberedEquation2.svg) |
where 
denotes the Kronecker product and 
is the 
identity matrix.
The eigenvalues of its coefficient matrix
are the pairwise sums of the eigenvalues of 
and 
, which gives the uniqueness criterion
(Higham 2002, pp. 305-319).
Replacing 
by 
gives the alternative form 
. Its homogeneous equation 
has a nonzero
solution iff 
and 
have a common eigenvalue (Sylvester
1884, Trott 2006). For square matrices of the same
size, an invertible solution exists iff
the two are similar matrices.
The Bartels-Stewart algorithm solves the equation after applying a Schur decomposition to 
and 
.
[F4]." Commun. ACM 15, 820-826, 1972.
." C. R. Acad. Sci. Paris 99, 67-71
and 115-116, 1884.