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

Abstract

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