Sylvester equation

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In control theory, the Sylvester equation is the matrix equation of the form

AX + XB = C,

where A,B,X,C are matrices.

Contents 1 Existence and uniqueness of the solution 2 Numerical solutions 3 See also 4 References 5 Notes

[edit] Existence and uniqueness of the solution

Using the Kronecker product notation and the vectorization operator , we can rewrite the equation in the form

where I_n is the identity matrix. In this form, the Sylvester equation can be seen as a linear system of dimension .^[1]

If A = ULU ^{− 1} and B^T = VMV ^{− 1} are the Jordan canonical forms of A and B^T, and λ_i and μ_j are their eigenvalues, one can write

Since is upper triangular with diagonal elements λ_i + μ_j, the matrix on the left hand side is singular if and only if there exist i and j such that λ_i = − μ_j.

Therefore, we have proved that the Sylvester equation has a unique solution if and only if A and − B have no common eigenvalues.

[edit] Numerical solutions

A classical algorithm for the numerical solution of the Sylvester equation is the Bartels--Stewart algorithm, which consists in transforming A and B into Schur form by a QR algorithm, and then solving the resulting triangular system via back-substitution. This algorithm, whose computational cost is O(n³) arithmetical operations, is used, among others, by LAPACK, Matlab and GNU Octave (in the syl function).

[edit] See also

• Lyapunov equation

[edit] References

R. H. Bartels and G. W. Stewart, Solution of the matrix equation $AX +XB = C$, Comm. ACM, 15 (1972), pp. 820 – 826.

[edit] Notes

1. ^ Rewriting the equation in this form is not advised for the numerical solution, though, since the linear system version is costly to solve and can be ill-conditioned