Note on a paper of Broyden

Recently, Broyden [Optim. Methods Software 8 (3-4) (1998) 185-199] proved a property of orthogonal matrices from which he derived Farkas' lemma and some related results. It is shown that Broyden's result straightforwardly follows from well-known theorems of the alternative, like Motzkin's transposition theorem and 'Tucker's theorem, which are all logically equivalent to Farkas' lemma; we also answer the question of Broyden on how to efficiently compute the sign matrix of an orthogonal matrix. Finally, we raise some related questions about possible generalizations of Broyden's result. (C) 1999 Elsevier Science B.V. All rights reserved.