Peano existence theorem

In mathematics, in the field of control theory, the Sylvester equation is a matrix equation of the form

$AX+XB=C,$

where $A,B,X,C$ are $n\times n$ matrices: $A,B,C$ are given and the problem is to find $X$.

Existence and uniqueness of the solutions

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 the syl function in that language.

See also

• Lyapunov equation
• Algebraic Riccati equation

References

• J. Sylvester, Sur l’equations en matrices $px=xq$, 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 $AX+XB=C$, Comm. ACM, 15 (1972), pp. 820 – 826.
• 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.
• 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.

External links

• Online solver for arbitrary sized matrices.
• Mathematica function to solve the Sylvester equation