Almost Commutative Matrices

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.