Powers of Random Matrices

Abstract

If we select an \(n \times n\) orthogonal matrix \(X\) "at random", using the uniform distribution on the orthogonal group \(\mathrm{O}(n)\), then the powers of \(X\) are not uniformly distributed in \(\mathrm{O}(n)\). However, as \(n\) increases, the distribution of \(X^n\) stabilizes. We prove this, consider generalizations to matrices in other compact Lie groups, and make some remarks about random matrices in other Lie groups.