Bibliography
P. Alexandroff undH. Hopf, Topologie I, Berlin (1935).
A. Brauer, A new proof of theorems of Perron and Frobenius on nonnegative matrices, Duke Math. Journ.,24, 367–378 (1957).
G. Debreu andI. N. Herstein, Nonnegative square matrices, Econometrica,21, 597–607 (1953).
E. Egerváry, On a lemma of Stieltjes on matrices, Acta Scient. Math. Szeged,15, 99–103 (1953–1954).
K. Fan, On systems of linear inequalities, Linear inequalities and related systems (Ann. of Math. Studies, No. 38), edited by H. W. Kuhn and A. W. Tucker, Princeton, 99–156 (1956).
M. Fréchet, Recherches thériques modernes sur le calcul des probabilités, Second livre, Paris (1938).
G. Frobenius, Über Matrizen aus positiven Elementen, Sitzungsber. Preuss. Akad. Wiss. Berlin, 471–476 (1908), 514–518 (1909).
G. Frobenius, Über Matrizen aus nicht negativen Elementen, Sitzungsber. Preuss. Akad. Wiss. Berlin, 456–477 (1912).
F. R. Gantmaher, Theory of matrices (Russian), Moscow (1954).
H. E. Goheen, On a lemma of Stieltjes on matrices, Amer. Math. Monthly,56, 328–329 (1949).
A. S. Householder, On the convergence of matrix iterations, Journ. Assoc. Computing Machinery,3, 314–324 (1956).
W. Hurewicz andH. Wallman, Dimension theory, Princeton (1941).
M. Janet, Sur les systèmes d'équations aux dérivées partielles. Journ. Math. pures et appl., (8e série),3, 65–151 (1920).
B. Knaster, C. Kuratowski undS. Mazurkiewicz, Ein Beweis des Fixpunktsatzes fürn-dimensionale Simplexe, Fund. Math.,14, 132–137 (1929).
A. Ostrowski, Über die Determinanten mit überwiegender Hauptdiagonale, Commentarii Math. Helvetici,10, 69–96 (1937).
A. Ostrouwski, Note on bounds for determinants with dominant principal diagonal, Proc. Amer. Math. Soc.,3, 26–30 (1952).
O. Perron, Zur Theorie der Matrices, Math. Annalen,64, 248–263 (1907).
G. B. Price, Bounds for determinants with dominant principal diagonal, Proc. Amer. Math. Soc.,2, 497–502 (1951).
H. Samelson, On the Perron-Frobenius theorem, Michigan Math. Journ.,4, 57–59 (1957).
H. Schneider, An inequality for latent roots applied to determinants with dominant principal diagonal, Journ. London Math. Soc.,28, 8–20 (1953).
E. Sperner, Neuer Beweis für die Invarianz der Dimensionszahl und des Gebietes, Abhandl. Math. Sem. Hamburg. Univ.,6, 265–272 (1928).
O. Taussky, A recurring theorem on determinants, Amer. Math. Monthly,56, 672–676 (1949).
J. L. Ullman, On a theorem of Frobenius, Michigan Math. Journ.,1, 189–193 (1952).
H. Wielandt, Unzerlegbare, nicht negative Matrizen, Math. Zeitschr.,52, 642–648 (1950).
Y. K. Wong, Some properties of the proper values of a matrix, Proc. Amer. Math. Soc.,6, 891–899 (1955).
Author information
Authors and Affiliations
Additional information
This paper was prepared while the author was working under a National Bureau of Standards contract with the American University.
Rights and permissions
About this article
Cite this article
Fan, K. Topological proofs for certain theorems on matrices with non-negative elements. Monatshefte für Mathematik 62, 219–237 (1958). https://doi.org/10.1007/BF01303967
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01303967