In mathematics, in the field of control theory, a Sylvester equation is a matrix equation of the form:[1] It is named after English mathematician James Joseph Sylvester.
All matrices are assumed to have coefficients in the complex numbers.
For the equation to make sense, the matrices must have appropriate sizes, for example they could all be square matrices of the same size.
But more generally, A and B must be square matrices of sizes n and m respectively, and then X and C both have n rows and m columns.
A Sylvester equation has a unique solution for X exactly when there are no common eigenvalues of A and −B.
More generally, the equation AX + XB = C has been considered as an equation of bounded operators on a (possibly infinite-dimensional) Banach space.
In this case, the condition for the uniqueness of a solution X is almost the same: There exists a unique solution X exactly when the spectra of A and −B are disjoint.
[2] Using the Kronecker product notation and the vectorization operator
, we can rewrite Sylvester's equation in the form where
In this form, the equation can be seen as a linear system of dimension
unknowns and the same number of equations.
admits only the trivial solution
be a solution to the abovementioned homogeneous equation.
denotes the spectrum of a matrix.
This proves the "if" part of the theorem.
is a nontrivial solution to the aforesaid homogeneous equation, justifying the "only if" part of the theorem.
As an alternative to the spectral mapping theorem, the nonsingularity of
in part (i) of the proof can also be demonstrated by the Bézout's identity for coprime polynomials.
The theorem remains true for real matrices with the caveat that one considers their complex eigenvalues.
satisfy the homogenous equation
The answer is that these two matrices are similar exactly when there exists a matrix X such that AX − XB = C. In other words, X is a solution to a Sylvester equation.
This is known as Roth's removal rule.
[4] One easily checks one direction: If AX − XB = C then Roth's removal rule does not generalize to infinite-dimensional bounded operators on a Banach space.
[5] Nevertheless, Roth's removal rule generalizes to the systems of Sylvester equations.
[6] A classical algorithm for the numerical solution of the Sylvester equation is the Bartels–Stewart algorithm, which consists of transforming
into Schur form by a QR algorithm, and then solving the resulting triangular system via back-substitution.
This algorithm, whose computational cost is
arithmetical operations,[citation needed] is used, among others, by LAPACK and the lyap function in GNU Octave.
[7] See also the sylvester function in that language.
[8][9] In some specific image processing applications, the derived Sylvester equation has a closed form solution.