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| In [[mathematics]], in the field of [[control theory]], the '''Sylvester equation''' is a [[Matrix (mathematics)|matrix]] [[equation]] of the form
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| :<math>A X + X B = C,</math>
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| where <math>A,B,X,C</math> are <math>n \times n</math> matrices: <math>A,B,C</math> are given and the problem is to find <math>X</math>.
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| ==Existence and uniqueness of the solutions==
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| Using the [[Kronecker product]] notation and the [[Vectorization (mathematics)|vectorization operator]] <math>\operatorname{vec}</math>, we can rewrite the equation in the form
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| :<math> (I_n \otimes A + B^T \otimes I_n) \operatorname{vec}X = \operatorname{vec}C,</math>
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| where <math>I_n</math> is the <math>n \times n</math> [[identity matrix]]. In this form, the Sylvester equation can be seen as a [[linear system]] of dimension <math>n^2 \times n^2</math>.<ref> 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]].</ref>
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| If <math>A=ULU^{-1}</math> and <math>B^T=VMV^{-1}</math> are the [[Jordan canonical form]]s of <math>A</math> and <math>B^T</math>, and <math>\lambda_i</math> and <math>\mu_j</math> are their [[eigenvalues]], one can write
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| :<math>I_n \otimes A + B^T \otimes I_n = (V\otimes U)(I_n \otimes L + M \otimes I_n)(V \otimes U)^{-1}.</math>
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| Since <math>(I_n \otimes L + M \otimes I_n)</math> is [[triangular matrix| upper triangular]] with diagonal elements <math>\lambda_i+\mu_j</math>, the matrix on the left hand side is singular if and only if there exist <math>i</math> and <math>j</math> such that <math>\lambda_i=-\mu_j</math>.
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| Therefore, we have proved that the Sylvester equation has a unique solution if and only if <math>A</math> and <math>-B</math> have no common eigenvalues.
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| ==Numerical solutions==
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| A classical algorithm for the numerical solution of the Sylvester equation is the Bartels–Stewart algorithm, which consists of transforming <math>A</math> and <math>B</math> into [[Schur decomposition|Schur form]] by a [[QR algorithm]], and then solving the resulting triangular system via [[Triangular matrix|back-substitution]]. This algorithm, whose computational cost is [[Big O notation|O]]<math>(n^3)</math> arithmetical operations, is used, among others, by [[LAPACK]] and the <code>lyap</code> function in [[GNU Octave]]. See also the <code>syl</code> function in that language.
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| ==See also==
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| * [[Lyapunov equation]]
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| * [[Algebraic Riccati equation]]
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| ==References==
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| * J. Sylvester, Sur l’equations en matrices <math>px = xq</math>, ''[[C. R. Acad. Sc. Paris]]'', 99 (1884), pp. 67 – 71, pp. 115 – 116.
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| * R. H. Bartels and G. W. Stewart, Solution of the matrix equation <math>AX +XB = C</math>, ''[[Comm. ACM]]'', 15 (1972), pp. 820 – 826.
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| * R. Bhatia and P. Rosenthal, How and why to solve the operator equation <math>AX -XB = Y </math> ?, ''[[Bull. London Math. Soc.]]'', 29 (1997), pp. 1 – 21.
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| * 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.
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| ==Notes==
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| <references/>
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| ==External links==
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| * [http://calculator-fx.com/calculator/linear-algebra/solve-sylvester-equation Online solver for arbitrary sized matrices.]
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| * [http://reference.wolfram.com/mathematica/ref/LyapunovSolve.html Mathematica function to solve the Sylvester equation]
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| [[Category:Matrices]]
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| [[Category:Control theory]]
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