Vol. 32, No. 1, 1970

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Matric polynomials which are higher commutators

Edmond Dale Dixon

Vol. 32 (1970), No. 1, 55–63

Let A be an n × n matrix defined over a field F of characteristic greater than n. For each n × n matrix X we define

          X1 = [A,X ]0 = X
Xh+1 = [A,X ]h = [A,Xh ] = AXh − XhA

for each positive integer h. Then X is defined to be k-commutative with A if and only if

[A,X ]k = 0, [A, X]k−1 ⁄= 0.

Let P(x) be a polynomial such that P(A)0. Specifically, assume that

       n∑−1   i
P(A ) =   λiA ⁄= 0

where p is a positive integer, each λi is a scalar from F, and λp0. In this paper we study, for each positive integer k, the matrices X such that

[A, X]k = P (A).

We specify a polynomial P(A) in the form (3) and show how the maximal value of k for which (4) has a solution depends on the polynomial P(A). In Theorem 3 it is assumed that A is nonderogatory. Since the only matrices which commute with A in this case are polynomials in A, we are, in effect, establishing a more precise bound for k in (2) by predetermining Xk.

In the derogatory case, a matrix which is not a polynomial in A may commute with A. However, Theorem 4 shows that if we choose a polynomial P(A) as Xk, then the maximal value of k depends on the polynomial P.

Mathematical Subject Classification
Primary: 15.35
Received: 6 December 1968
Published: 1 January 1970
Edmond Dale Dixon