Vol. 37, No. 3, 1971

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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

Let A be an n-square matrix over a field F of characteristic 0. The additive commutator operator defined by A, TAX = 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 TA is always odd and at least

2[m + E + (k − 2)e− k]+ 1

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
Received: 4 June 1970
Published: 1 June 1971
Mohammad Shafqat Ali
Marvin David Marcus