Let L be an invertible linear
map on the space M(n,k) of n-square matrices over a field k of characteristic not 2.
In this paper we classify all such L which preserve a particular orthogonal group of a
nonsingular symmetric bilinear form. We use some elementary facts about algebraic
groups and an idea of Dieudonné’s. There is some indication that our use of
an algebraic geometric setting is the proper one for many problems of this
type.
There are a considerable number of results which may be paraphrased as
follows: “let L : M(n,k) → M(n,k) be a linear transformation that preserves
some property related to matrix multiplication. Then L is almost an inner
automorphism of M(n,k).” While the statement of these results exhibits a
large degree of similarity, an examination of the proofs reveals almost no
similarity. The property in question could be determinant, nonsingularity, or
orthogonality.
