Applications of the Gordan–Stiemke Theorem in Combinatorial Matrix Theory
B. David Saunders, Hans Schneider · SIAM Review · 1979
By use of the Gordan–Stiemke Theorem of the alternative we demonstrate the similarity of four theorems in combinatorial matrix theory. Each theorem contains five equivalent conditions, one of which is the existence in a given pattern of a line-sum-symmetric or constant-line-sum matrix which is semi-positive or strictly positive for the pattern. A generalization of the Gordan–Stiemke Theorem is stated in terms of complementary faces of the positive orthant and combinatorial applications are given. Many of our results are classical, but some may be new.