On the degree of the minimal polynomial of a commutator operator

Mohammad Shafqat Ali and Marvin David Marcus

Vol. 37 (1971), No. 3, 561–565

DOI: 10.2140/pjm.1971.37.561

Abstract

Let A be an n-square matrix over a field F of characteristic 0. The additive commutator operator defined by A, T_AX = AX − XA, can be regarded as a linear transformation on the space of all n-square matrices X over F. Following earlier papers by 0. Taussky and H. Wielandt and one of the present authors, we show that the degree of the minimal polynomial of T_A is always odd and at least

where m is the degree of the minimal polynomial of A,k is the number of distinct eigenvalues of A, and E(e) is the largest (least) integer among the degrees of the distinct highest degree elementary divisors of the characteristic matrix of A.

Mathematical Subject Classification 2000

Primary: 15A24

Milestones

Received: 4 June 1970

Published: 1 June 1971