CONSISTENCY AND EFFICIENT SOLUTION OF THE SYLVESTER EQUATION FOR ⋆-CONGRUENCE
CONSISTENCY AND EFFICIENT SOLUTION OF THE SYLVESTER EQUATION FOR ⋆-CONGRUENCE
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DOI：
10.13001/1081-3810.1479
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发表时间：
2011
影响因子：
0.7
通讯作者：
Fernando De Ter;An M. Dopico
中科院分区：
文献类型：
--
作者：
Fernando De Ter;An M. Dopico
本文考虑矩阵方程 AX + X⋆B = C，其中矩阵 A 和 B 的大小分别为 m × n 和 n × m，未知 X 的大小为 n × m，运算符 (·) ⋆ 表示矩阵的转置或共轭转置。本文的第一部分回顾了解的存在性和唯一性的充要条件。这些条件先前由 Wimmer（H.K. Wimmer。具有对称约束的矩阵方程的 Roth 定理。线性代数应用，199:357-362, 1994.）、Byers 和 Kressner（R. Byers 和 D. Kressner。不变子空间的结构化条件数。SIAM J. Matrix Anal. Appl.， 28:326-347, 2006.)，作者为 Kressner、Schroder 和 Watkins（D. Kressner、C. Schroder 和 D.S. Watkins。用于回文甚至特征值问题的隐式 QR 算法。Numer. Algorithms, 51:209-238, 2009。）。这篇综述将Wimmer最初证明的复数域的存在条件推广到了两个不同特征的域。在第二部分中，开发了一种算法，在实数或复数平方情况下 m = n，当解唯一时，可以在 O(n3) 次失败中求解方程。该算法基于矩阵铅笔 A λB ⋆ 的广义 Schur 分解。方程 AX + X ⋆ B = C 与回文特征值问题相关，因此，平方复数情况最近引起了几位作者的注意。
In this paper, the matrix equation AX + X⋆B = C is considered, where the matrices A and B have sizes m × n and n × m, respectively, the size of the unknown X is n × m, and the operator (·) ⋆ denotes either the transpose or the conjugate transpose of a matrix. In the first part of the paper, necessary and sufficient conditions for the existence and uniqueness of solutions are reviewed. These conditions were obtained previously by Wimmer (H.K. Wimmer. Roth's theorems for matrix equations with symmetry constraints. Linear Algebra Appl., 199:357- 362, 1994.), by Byers and Kressner (R. Byers and D. Kressner. Structured condition numbers for invariant subspaces. SIAM J. Matrix Anal. Appl., 28:326-347, 2006.), and by Kressner, Schroder and Watkins (D. Kressner, C. Schroder, and D.S. Watkins. Implicit QR algorithms for palindromic and even eigenvalue problems. Numer. Algorithms, 51:209-238, 2009.). This review generalizes to fields of characteristic different from two the existence condition that Wimmer originally proved for the complex field. In the second part, an algorithm is developed, in the real or complex square case m = n, to solve the equation in O(n3) flops when the solution is unique. This algorithm is based on the generalized Schur decomposition of the matrix pencil A λB ⋆ . The equation AX + X ⋆ B = C is connected with palindromic eigenvalue problems and, as a consequence, the square complex case has attracted recently the attention of several authors.