The solution of the equation AX+X★B=0

• Published in: Linear algebra and its applications Vol. 438; no. 7; pp. 2817 - 2860
• Main Authors: De Terán, Fernando, Dopico, Froilán M., Guillery, Nathan, Montealegre, Daniel, Reyes, Nicolás
• Format: Journal Article
• Language: English
• Published: Elsevier Inc 01.04.2013
• Subjects: Kronecker canonical form; Matrix equations; Matrix pencils; Palindromic eigenvalue problems; Sylvester equation for ★congruence; Palindromic eigenvalue problems; 15A24; Sylvester equation for ★congruence; 15A21; 65F15; Matrix equations; Kronecker canonical form; Matrix pencils
• ISSN: 0024-3795; 1873-1856
• DOI: 10.1016/j.laa.2012.11.014

We describe how to find the general solution of the matrix equation AX+X★B=0, where A∈Cm×n and B∈Cn×m are arbitrary matrices, X∈Cn×m is the unknown, and X★ 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+λB★ 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+λB★ 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+λB★. The general solution of the homogeneous equation AX+X★B=0 is essential to finding the general solution of AX+X★B=C, which is related to palindromic eigenvalue problems that have attracted considerable attention recently.