Thompson was awarded a Ph.D. in 1960 for his thesis Commutators in the Special and General Linear Groups. In 1961 Thompson published a paper Commutators in the special and general linear groups in the Transactions of the American Mathematical Society which was based on results from his thesis. In the paper he looks at matrices over a field K with at least 4 elements. He proves that a matrix A in SL(n, K) can be written in the form B-1C-1BC, where B, C belong to SL(n, K), except perhaps when A is a scalar matrix and K has characteristic 0. A scalar matrix in SL(n, K) is, however, a product of two commutators. Although the paper does not examine the cases where K = GF(2) or GF(3), Thompson gave complete information about these in his thesis and went on to publish the results for K = GF(3) in On matrix commutators (1962) and the case for K = GF(2) in Commutators of matrices with coefficients from the field of two elements (1962) published in the Duke Mathematical Journal. These papers are described by Charles R Johnson and Morris Newman in the following way:-
In that very detailed work, Thompson answered nearly all the major questions in the subject and revealed what would become a hallmark of his work: a willingness and an ability to make unusually elaborate algebraic calculations in order to answer a question. It was not that he did not appreciate external or efficient, implicit tools if they were available. Quite the contrary: he was a major proponent of the use of other parts of mathematics that could be useful in matrix theory.After Thompson was awarded his doctorate he returned to Canada, going back to the University of British Columbia where he was on the faculty for three years. In 1963 he moved to the University of California at Santa Barbara where he remained on the faculty for the rest of his career.
Thompson published over 120 papers and four undergraduate textbooks during his career. He attacked problems over a long period with great persistence and emphasis on detail. As a result he published a number of series of papers attacking particular problems. For example, he published nine papers on Principal submatrices in which he conducted a deep investigation of interlacing properties relating the eigenvalues of a matrix to the eigenvalues of a principal submatrix. For example Invariant factors of integral quaternion matrices which Thompson published in Linear and Multilinear Algebra in 1987 was reviewed by Morris Newman as follows:-
[Thompson's] purpose in this paper is to develop the theory of unimodular equivalence for matrices whose entries come from the Hurwitz ring of integral quaternions. This is done in complete detail, and a normal form is obtained which exhibits as much uniqueness as is possible for the case of a noncommutative ring. The author also proves a number of results concerned with the relationship between the invariant factors of a full matrix and the invariant factors of a submatrix, which generalize his earlier work on this subject.In 1990 Thomson spoke at a conference in Auburn, Alabama. His lecture High, low, and quantitative roads in linear algebra. Directions in matrix theory was a particularly interesting one for in it he attempted to predict the future direction of core linear algebra. He discussed: quantitative prediction; high and low roads; the numerical range; similarity invariants of principal submatrices; commutators; the triangle inequality; the facial structure of the unit ball; the Gershgorin circle theorem; matrices, graphs, inertia, number theory; power embeddings and dilatations; the Schubert calculus; the spectrum of a sum of Hermitian matrices; the Hadamard-Schur product; the exponential function; the exponential function and commutativity; integral quadratic forms; the matrix-valued numerical range; inequalities with subtracted terms; and further uses of the computer.
Editorial work was important to Thompson: he was a contributing editor of Linear Algebra and its Applications and an editor of the Society for Industrial and Applied Mathematics Journal on Matrix Analysis. Thompson's influence on linear algebra was felt in other ways too for he was one of the founders of the International Linear Algebra Society and of the journal Linear and Multilinear Algebra. Given this experience his article Author vs. referee: a case history for middle level mathematicians which appeared in the American Mathematical Monthly in 1983 is all the more interesting. He explains in the introduction why he wrote it:-
This note evolved from a referee's rejection of a research paper that I wrote. The reasoning behind the rejection was perhaps unusual, and leads to a not altogether trivial question concerning the role of the referee in the professional development of a mathematician. The discussion will be more candid than is customary, and this may add spice to the article, since confession of failure, or even of sin, is always interesting.Thompson's health deteriorated and heart problems were diagnosed:-
He retained his sense of humour and optimism to the end; although fully aware of his condition, he did not let it interfere with his work or discourage him. In fact, shortly before his death, he mentioned that he had told his doctors they were obligated to keep him alive for at least four more years, since he had just purchased an athletic club membership for that amount of time.Sadly the doctors failed to give him four years for he died while waiting for a heart transplant. Sadly the award of the International Linear Algebra Society's Hans Schneider Prize in Linear Algebra came after his death.
Article by: J J O'Connor and E F Robertson