The solution of the equation AX plus X☆B=0

被引：25

作者：

De Teran, Fernando ^{[1
]}

Dopico, Froilan M. ^{[1
,2
]}

Guillery, Nathan ^{[3
]}

Montealegre, Daniel ^{[4
]}

Reyes, Nicolas ^{[5
]}

机构：

[1] Univ Carlos III Madrid, Dept Matemat, Leganes 28911, Spain

[2] Univ Carlos III Madrid, Inst Ciencias Matemat CSIC UAM UCM UC3M, Leganes 28911, Spain

[3] Univ Calif Santa Barbara, Santa Barbara, CA 93106 USA

[4] Univ Calif Los Angeles, Los Angeles, CA 90024 USA

[5] Univ Massachusetts, Amherst, MA 01003 USA

来源：

LINEAR ALGEBRA AND ITS APPLICATIONS | 2013年 / 438卷 / 07期

关键词：

Kronecker canonical form; Matrix equations; Matrix pencils; Palindromic eigenvalue problems; Sylvester equation for (star)congruence; GENERALIZED SCHUR DECOMPOSITION; ARBITRARY PENCIL-A; ASTERISK-CONGRUENCE; EIGENVALUE PROBLEMS; ROBUST SOFTWARE; ERROR-BOUNDS; LAMBDA-B; MATRIX; ALGORITHMS; FORMS;

D O I：

10.1016/j.laa.2012.11.014

中图分类号：

O29 [应用数学];

学科分类号：

070104 ;

摘要：

We describe how to find the general solution of the matrix equation AX + (XB)-B-star = 0, where A is an element of C-mxn and B is an element of C-nxm are arbitrary matrices, X is an element of C-nxm is the unknown, and X-star denotes either the transpose or the conjugate transpose of X. We first show that the solution can be obtained in terms of the Kronecker canonical form of the matrix pencil A + lambda B-star and the two nonsingular matrices which transform this pencil into its Kronecker canonical form. We also give a complete description of the solution provided that these two matrices and the Kronecker canonical form of A + lambda B-star are known. As a consequence, we determine the dimension of the solution space of the equation in terms of the sizes of the blocks appearing in the Kronecker canonical form of A + lambda B-star. The general solution of the homogeneous equation AX + (XB)-B-star = 0 is essential to finding the general solution of AX + (XB)-B-star = C, which is related to palindromic eigenvalue problems that have attracted considerable attention recently. (C) 2012 Elsevier Inc. All rights reserved.

引用

页码：2817 / 2860

页数：44

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