Peano existence theorem: Difference between revisions
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In [[mathematics]], in the field of [[control theory]], the '''Sylvester equation''' is a [[Matrix (mathematics)|matrix]] [[equation]] of the form | |||
:<math>A X + X B = C,</math> | |||
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>. | |||
==Existence and uniqueness of the solutions== | |||
Using the [[Kronecker product]] notation and the [[Vectorization (mathematics)|vectorization operator]] <math>\operatorname{vec}</math>, we can rewrite the equation in the form | |||
:<math> (I_n \otimes A + B^T \otimes I_n) \operatorname{vec}X = \operatorname{vec}C,</math> | |||
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> | |||
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 | |||
:<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> | |||
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>. | |||
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. | |||
==Numerical solutions== | |||
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. | |||
==See also== | |||
* [[Lyapunov equation]] | |||
* [[Algebraic Riccati equation]] | |||
==References== | |||
* J. Sylvester, Sur l’equations en matrices <math>px = xq</math>, ''[[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 <math>AX +XB = C</math>, ''[[Comm. ACM]]'', 15 (1972), pp. 820 – 826. | |||
* 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. | |||
* 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. | |||
==Notes== | |||
<references/> | |||
==External links== | |||
* [http://calculator-fx.com/calculator/linear-algebra/solve-sylvester-equation Online solver for arbitrary sized matrices.] | |||
* [http://reference.wolfram.com/mathematica/ref/LyapunovSolve.html Mathematica function to solve the Sylvester equation] | |||
[[Category:Matrices]] | |||
[[Category:Control theory]] |
Revision as of 12:25, 3 February 2014
In mathematics, in the field of control theory, the Sylvester equation is a matrix equation of the form
where are matrices: are given and the problem is to find .
Existence and uniqueness of the solutions
Using the Kronecker product notation and the vectorization operator , we can rewrite the equation in the form
where is the identity matrix. In this form, the Sylvester equation can be seen as a linear system of dimension .[1]
If and are the Jordan canonical forms of and , and and are their eigenvalues, one can write
Since is upper triangular with diagonal elements , the matrix on the left hand side is singular if and only if there exist and such that .
Therefore, we have proved that the Sylvester equation has a unique solution if and only if and 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 and into Schur form by a QR algorithm, and then solving the resulting triangular system via back-substitution. This algorithm, whose computational cost is O 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
References
- J. Sylvester, Sur l’equations en matrices , 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 , Comm. ACM, 15 (1972), pp. 820 – 826.
- R. Bhatia and P. Rosenthal, How and why to solve the operator equation ?, 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.
Notes
- ↑ 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.