In mathematics, in the field of control theory, a Sylvester equation is a matrix equation of the form:
A X + X B = C . {\displaystyle AX+XB=C.}
It is named after English mathematician James Joseph Sylvester. Then given matrices A, B, and C, the problem is to find the possible matrices X that obey this equation. All matrices are assumed to have coefficients in the complex numbers. For the equation to make sense, the matrices must have appropriate sizes, for example they could all be square matrices of the same size. But more generally, A and B must be square matrices of sizes n and m respectively, and then X and C both have n rows and m columns. A Sylvester equation has a unique solution for X exactly when there are no common eigenvalues of A and −B. More generally, the equation AX + XB = C has been considered as an equation of bounded operators on a (possibly infinite-dimensional) Banach space. In this case, the condition for the uniqueness of a solution X is almost the same: There exists a unique solution X exactly when the spectra of A and −B are disjoint.
Existence and uniqueness of the solutions Using the Kronecker product notation and the vectorization operator vec {\displaystyle \operatorname {vec} } , we can rewrite Sylvester's equation in the form
( I m ⊗ A + B T ⊗ I n ) vec X = vec C , {\displaystyle (I_{m}\otimes A+B^{T}\otimes I_{n})\operatorname {vec} X=\operatorname {vec} C,}
where A {\displaystyle A} is of dimension n × n {\displaystyle n\!\times \!n} , B {\displaystyle B} is of dimension m × m {\displaystyle m\!\times \!m} , X {\displaystyle X} of dimension n × m {\displaystyle n\!\times \!m} and I k {\displaystyle I_{k}} is the k × k {\displaystyle k\times k} identity matrix. In this form, the equation can be seen as a linear system of dimension m n × m n {\displaystyle mn\times mn} . Theorem. Given matrices A ∈ C n × n {\displaystyle A\in \mathbb {C} ^{n\times n}} and B ∈ C m × m {\displaystyle B\in \mathbb {C} ^{m\times m}} , the Sylvester equation A X + X B = C {\displaystyle AX+XB=C} has a unique solution X ∈ C n × m {\displaystyle X\in \mathbb {C} ^{n\times m}} for any C ∈ C n × m {\displaystyle C\in \mathbb {C} ^{n\times m}} if and only if A {\displaystyle A} and − B {\displaystyle -B} do not share any eigenvalue. Proof. The equation A X + X B = C {\displaystyle AX+XB=C} is a linear system with m n {\displaystyle mn} unknowns and the same number of equations. Hence it is uniquely solvable for any given C {\displaystyle C} if and only if the homogeneous equation
A X + X B = 0 {\displaystyle AX+XB=0}
admits only the trivial solution 0 {\displaystyle 0} . (i) Assume that A {\displaystyle A} and − B {\displaystyle -B} do not share any eigenvalue. Let X {\displaystyle X} be a solution to the abovementioned homogeneous equation. Then A X = X ( − B ) {\displaystyle AX=X(-B)} , which can be lifted to
A k X = X ( − B ) k {\displaystyle A^{k}X=X(-B)^{k}}
for each k ≥ 0 {\displaystyle k\geq 0}
by mathematical induction. Consequently,
p ( A ) X = X p ( − B ) {\displaystyle p(A)X=Xp(-B)}
for any polynomial p {\displaystyle p} . In particular, let p {\displaystyle p} be the characteristic polynomial of A {\displaystyle A} . Then
p ( A ) = 0 {\displaystyle p(A)=0} due to the Cayley–Hamilton theorem; meanwhile, the spectral mapping theorem tells us
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