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第1行： − In [[mathematics]], in the field of [[control theory]], a '''Sylvester equation''' is a [[Matrix (mathematics)|matrix]] [[equation]] of the form:<ref>This equation is also commonly written in the equivalent form of ''AX'' − ''XB'' = ''C''.</ref>+ − − 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 number]]s. 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. − In mathematics, in the field of control theory, a Sylvester equation is a matrix equation of the form:This equation is also commonly written in the equivalent form of AX − XB = C.+ − :A X + X B = C. − 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. − 在数学中，在控制论领域中，Sylvester 方程是一个矩阵方程的形式: 这个方程也通常以 AX-XB = c 的等价形式写成: a x + x b = c。然后给定矩阵 a、 b 和 c，问题是找到服从这个方程的可能矩阵 x。假定所有的矩阵都有复数系数。为了使方程有意义，矩阵必须有适当的大小，例如它们可以都是相同大小的方阵。但更一般地说，a 和 b 必须分别是大小为 n 和 m 的方阵，然后 x 和 c 都有 n 行和 m 列。+ − 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 operator]]s 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 [[Spectrum (functional analysis)|spectra]] of ''A'' and −''B'' are [[Disjoint sets|disjoint]].<ref>Bhatia and Rosenthal, 1997</ref> − − 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.Bhatia and Rosenthal, 1997 − − 当 a 和-b 没有共同的本征值时，Sylvester 方程对 x 有唯一的解。更一般地，方程 AX + XB = c 被认为是一个有界算子方程在一个(可能是无限维) Banach 空间上。在这种情况下，解 x 唯一的条件几乎相同: 当 a 和-b 的谱不相交时，存在唯一的解 x。巴蒂亚和罗森塔尔，1997 第145行：
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此词条暂由彩云小译翻译，翻译字数共974，未经人工整理和审校，带来阅读不便，请见谅。
'''西尔韦斯特方程'''是控制论中的矩阵方程，形式如下<ref>This equation is also commonly written in the equivalent form of ''AX'' − ''XB'' = ''C''.</ref>:
:<math>A X + X B = C.</math>
<math>A X + X B = C.</math>
其中''A''，''B''，''C''为已知的矩阵，问题是要找到能够满足方程的矩阵X。所有矩阵的系数都是复数。为了使方程有意义，矩阵的行和列需要满足一定条件，''A'' 和 ''B'' 都要是方阵，大小分别是''n''和''m''，而''X''和''C''要是''n''行''m''列的矩阵，''n''和''m''也可以相等，四个矩阵都是大小相同的方阵。
当且仅当 ''A'' 和-''b'' 没有共同的本征值时，西尔韦斯特有唯一解 ''X'' 。更一般地，方程 AX + XB = c 也可以视为(可能无限维中)巴拿赫空间中有界算子的方程。在这种情况下，此情形下，有唯一解 ''X'' 的充分必要条件几乎相同: ''A'' 和-''B'' 的谱不相交<ref>Bhatia and Rosenthal, 1997</ref>。
==Existence and uniqueness of the solutions==
==Existence and uniqueness of the 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 \mathcal{O}(n^3) arithmetical operations, is used, among others, by LAPACK and the lyap function in GNU Octave. See also the sylvester function in that language.The <code>syl</code> command is deprecated since GNU Octave Version 4.0 In some specific image processing application, the derived Sylvester equation has a closed form solution.
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 \mathcal{O}(n^3) arithmetical operations, is used, among others, by LAPACK and the lyap function in GNU Octave. See also the sylvester function in that language.The <code>syl</code> command is deprecated since GNU Octave Version 4.0 In some specific image processing application, the derived Sylvester equation has a closed form solution.
Sylvester 方程数值解的一个经典算法是 Bartels-Stewart 算法，该算法通过 QR 算法将 a 和 b 转化为 Schur 形式，然后通过反代换求解得到三角形方程组。该算法的计算代价是数学{ o }(n ^ 3)算术运算，其中包括 LAPACK ack 和 GNU Octave 中的 lyap 函数。请参阅该语言中的 sylvester 函数。自 GNU Octave Version 4.0以来，不推荐使用 < code > syl </code > 命令。在某些特定的图像处理应用程序中，导出的 Sylvester 方程有一个封闭的解。
Sylvester 方程数值解的一个经典算法是 Bartels-Stewart 算法，该算法通过 QR 算法将 a 和 b 转化为 Schur 形式，然后通过反代换求解得到三角形方程组。该算法的计算代价是数学{ o }(n ^ 3)算术运算，其中包括 LAPACK ack 和 GNU Octave 中的 lyap 函数。请参阅该语言中的 sylvester 函数。自 GNU Octave Version 4.0以来，不推荐使用 < code > syl 命令。在某些特定的图像处理应用程序中，导出的 Sylvester 方程有一个封闭的解。
==See also==
==See also==