Almost commutative matrices

Abstract

The n × n matrices A and X over a field F are called almost commutative if AX - XA = I . This equation cannot hold if the characteristic of F is either zero or greater than n . In the case where the characteristic of F divides n , certain pairs A and X , exist. It is the purpose of this paper not only to prove the existence of such pairs, but to construct (in terms of arbitrary parameters) the most general matrix X for a given matrix A . The methods use the rational canonical form so as to facilitate constructability.