The solution of the equation and its application to the theory of orbits

✍ Scribed by Fernando De Terán; Froilán M. Dopico

• Publisher: Elsevier Science
• Year: 2011
• Tongue: English
• Weight: 348 KB
• Volume: 434
• Category: Article
• ISSN: 0024-3795
• DOI: 10.1016/j.laa.2010.08.005

✦ Synopsis

We describe how to find the general solution of the matrix equation XA + AX T = 0, with A ∈ C n×n , which allows us to determine the dimension of its solution space. This result has immediate applications in the theory of congruence orbits of matrices in C n×n , because the set {XA + AX T : X ∈ C n×n } is the tangent space at A to the congruence orbit of A. Hence, the codimension of this orbit is precisely the dimension of the solution space of XA + AX T = 0. As a consequence, we also determine the generic canonical structure of matrices under the action of congruence. All these results can be directly extended to palindromic pencils A + λA T .