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.

Let A be an n X n matrix defined over a field F of characteristic greater than n. For each n x n matrix X we define (1) X 1 = [A,X]o = X X h+ί = [A, X] h = [A, X,] = AX h-X h A for each positive integer h. Then X is defined to be A>commutative with A if and only if (2) [A,X] k = 0, [A, JΓU-x * 0. Let P(x) be a polynomial such that P{A) Φ 0. Specifically, assume that (3) P(A) =2*^*0

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It is the purpose of this paper to find in terms of parameters the most generai matrix A'which is tf-commutative with respect to a given matrix A. The proofs will yield a method of rational construction for such a matrix X.

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