Sylvester equation

The Sylvester equation, commonly encountered in control theory, is the matrix equation of the form

$A X + X B = C,$

where $A,B,X,C$ are $n \times n$ matrices. $A,B,C$ are known. The problem is to find $X$.

Existence and uniqueness of the solution

Using the Kronecker product notation and the vectorization operator $\operatorname{vec}$, we can rewrite the equation in the form

$(I_n \otimes A + B^T \otimes I_n) \operatorname{vec}X = \operatorname{vec}C,$

where $I_n$ is the $n \times n$ identity matrix. In this form, the Sylvester equation can be seen as a linear system of dimension $n^2 \times n^2$.

If $A=ULU^{-1}$ and $B^T=VMV^{-1}$ are the Jordan canonical forms of $A$ and $B^T$, and $\lambda_i$ and $\mu_j$ are their eigenvalues, one can write

$I_n \otimes A + B^T \otimes I_n = (V\otimes U)(I_n \otimes L + M \otimes I_n)(V \otimes U)^{-1}.$

Since $(I_n \otimes L + M \otimes I_n)$ is upper triangular with diagonal elements $\lambda_i+\mu_j$, the matrix on the left hand side is singular if and only if there exist $i$ and $j$ such that $\lambda_i=-\mu_j$.

Therefore, we have proved that the Sylvester equation has a unique solution if and only if $A$ and $-B$ 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 $A$ 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$(n^3)$ arithmetical operations, is used, among others, by LAPACK and the lyap function in GNU Octave.

See also

• Lyapunov equation
• Algebraic Riccati equation

References

1. J. Sylvester, Sur l’equations en matrices $px = xq$, C.R. Acad. Sci. Paris, 99 (1884), pp. 67 – 71, pp. 115 – 116.

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

3. R. Bhatia and P. Rosenthal, How and why to solve the operator equation $AX -XB = Y$ ?, Bull. London Math. Soc., 29 (1997), pp. 1 – 21.

4. 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.