Note on a paper of Broyden

Recently, Broyden 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.