Inner derivations of division rings and canonical Jordan form of triangular operators

Abstract:

Let $D$ be a division ring and $k$ its center. We show that a generalized canonical Jordan form exists for triangularizable matrices $A$ over $D$ which are algebraic over $k$, i.e, satisfy $f(A) = 0$ for some nonzero polynomial $f$ over $k$. This canonical form is a direct sum of generalized Jordan blocks ${J_m}(\alpha ,\beta )$. This block is an $m$ by $m$ matrix whose diagonal entries are equal to $\alpha$, those on the first superdiagonal are equal to $\beta$, and all other entries are equal to zero. If $\alpha$ is separable over $k$ then we can choose $\beta = 1$, but in general this cannot be done.