Sylvester equation

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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

Notes

  1. 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.

External links