# Sylvester equation

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 .^{[1]}

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.

## Numerical solutions

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.

## See also

## References

- 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.

## Notes

- ↑ 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.