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 so me related results. It is shown that Broyden's result straightforwardly fol lows from well-known theorems of the alternative, like Motzkin's transposit ion theorem and 'Tucker's theorem, which are all logically equivalent to Fa rkas' lemma; we also answer the question of Broyden on how to efficiently c ompute the sign matrix of an orthogonal matrix. Finally, we raise some rela ted questions about possible generalizations of Broyden's result. (C) 1999 Elsevier Science B.V. All rights reserved.