In mathematics, in the field of control theory, the Sylvester equation is a matrix equation of the form
where are matrices: are given and the problem is to find .
Existence and uniqueness of the solutions
Using the Kronecker product notation and the vectorization operator , we can rewrite the equation in the form
where is the identity matrix. In this form, the Sylvester equation can be seen as a linear system of dimension .
If and are the Jordan canonical forms of and , and and are their eigenvalues, one can write
Since is upper triangular with diagonal elements , the matrix on the left hand side is singular if and only if there exist and such that .
Therefore, we have proved that the Sylvester equation has a unique solution if and only if and have no common eigenvalues.
A classical algorithm for the numerical solution of the Sylvester equation is the Bartels–Stewart algorithm, which consists of transforming and into Schur form by a QR algorithm, and then solving the resulting triangular system via back-substitution. This algorithm, whose computational cost is O arithmetical operations, is used, among others, by LAPACK and the
lyap function in GNU Octave. See also the
syl function in that language.
- J. Sylvester, Sur l’equations en matrices , C. R. Acad. Sc. Paris, 99 (1884), pp. 67 – 71, pp. 115 – 116.
- R. H. Bartels and G. W. Stewart, Solution of the matrix equation , Comm. ACM, 15 (1972), pp. 820 – 826.
- R. Bhatia and P. Rosenthal, How and why to solve the operator equation ?, Bull. London Math. Soc., 29 (1997), pp. 1 – 21.
- S.-G. Lee and Q.-P. Vu, Simultaneous solutions of Sylvester equations and idempotent matrices separating the joint spectrum, Linear Algebra and its Applications, 435 (2011), pp. 2097 – 2109.
- ↑ However, rewriting the equation in this form is not advised for the numerical solution since this version is costly to solve and can be ill-conditioned.