On the orbit of the centralizer of a matrix
Volume 92 / 2002
Colloquium Mathematicum 92 (2002), 243-255
MSC: 47A15, 15A21.
DOI: 10.4064/cm92-2-8
Abstract
Let $A$ be a complex $n \times n$ matrix. Let $\{ A \}'$ be its commutant in $M_n({\mathbb C})$, and $C(A)$ be its centralizer in ${\rm GL}(n, {\mathbb C})$. Consider the standard $C(A)$-action on ${\mathbb C}^n$. We describe the $C(A)$-orbits via invariant subspaces of $\{ A \}'$. For example, we count the number of $C(A)$-orbits as well as that of invariant subspaces of $\{ A \}'$.
