Conjugation orbits in the square matrices - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T06:59:32Z http://mathoverflow.net/feeds/question/57924 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/57924/conjugation-orbits-in-the-square-matrices Conjugation orbits in the square matrices Marc Palm 2011-03-09T07:04:01Z 2011-03-09T12:05:14Z <p>Consider the action of $GL_n(R)$ on $M_{n \times n}(R)$ by conjugation, where $R$ is a ring (or field)? How can we classify the orbits? To what extent does the characteristic polynomial and the trace classify the orbits?</p> http://mathoverflow.net/questions/57924/conjugation-orbits-in-the-square-matrices/57928#57928 Answer by Amritanshu Prasad for Conjugation orbits in the square matrices Amritanshu Prasad 2011-03-09T07:37:00Z 2011-03-09T07:37:00Z <p>For a field, this is given by the rational canonical form (see Section 7.2 of Hoffman and Kunze's Linear Algebra, for example). Even in this case, the trace and characteristic polynomial are quite weak as invariants. What you need are the <em>invariant factors</em>. For general rings, this is very hard and often a <em>wild classifcation problem</em>. For rings like $\mathbf Z/p^k\mathbf Z$ look at <a href="http://dx.doi.org/10.1080/00927870902747266" rel="nofollow">this paper</a>.</p>