Vol. 20, No. 2, 1967

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ISSN: 0030-8730
On a class of matrix equations

Robert Charles Thompson

Vol. 20 (1967), No. 2, 289–316
Abstract

Let K be a field of characteristic p,p zero or prime, and let L be the algebraic closure of K. Let Mn(K) denote the matrix algebra of n-square matrices with elements in K. The commutator of A,B Mn(K) is defined by (A,B) = AB BA. It is the object of this paper to examine the following two questions.

I. Given exactly one of the three matrices A,B,C Mn(K), to determine necessary and sufficient conditions in order that the other two matrices will exist in Mn(K) such that

C = (A,B ),(C,A ) = 0,C ⁄= 0.
(1)

II. Given exactly one of the three matrices A,B,C Mn(K), to determine necessary and sufficient conditions in order that the other two matrices will exist in Mn(K) such that

C = (A,B ),(C,A ) = 0,(C,B) = 0,C ⁄= 0.
(2)

We shall obtain complete solutions to all these problems, except that, in Question I when C is the given matrix and 0 < p n, we obtain only a partial solution. As a consequence of our results, we are able to find conditions that are sufficient, and sometimes necessary and sufficient, in order that solutions exist in Mn(K) for certain complicated families of commutator equations related to (1) or (2).

Mathematical Subject Classification
Primary: 15.35
Secondary: 16.48
Milestones
Received: 2 December 1965
Published: 1 February 1967
Authors
Robert Charles Thompson