The Structure of Powers of Nonnegative Matrices I. The Index of Convergence
B. R. Heap, M. Stuart Lynn · SIAM Journal on Applied Mathematics · 1966
Previous article The Structure of Powers of Nonnegative Matrices I. The Index of ConvergenceB. R. Heap and M. S. LynnB. R. Heap and M. S. Lynnhttps://doi.org/10.1137/0114052PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] Claude Berge, The theory of graphs and its applications, Translated by Alison Doig, Methuen & Co. Ltd., London, 1962x+247 MR0132541 (24:A2381) Google Scholar[2] Alfred Brauer and , James E. Shockley, On a problem of Frobenius, J. Reine Angew. Math., 211 (1962), 215–220 MR0148606 (26:6113) Google Scholar[3] A. L. Dulmage and , N. S. Mendelsohn, The characteristic equation of an imprimitive matrix, J. Soc. Indust. Appl. Math., 11 (1963), 1034–1045 10.1137/0111076 MR0162812 (29:116) LinkISIGoogle Scholar[4] A. L. Dulmage and , N. S. Mendelsohn, Gaps in the exponent set of primitive matrices, Illinois J. Math., 8 (1964), 642–656 MR0181645 (31:5872) CrossrefGoogle Scholar[5] A. L. Dulmage and , N. S. Mendelsohn, The structure of powers of non-negative matrices, Canad. J. Math., 17 (1965), 318–330 MR0213384 (35:4248) CrossrefISIGoogle Scholar[6] G. Frobenius, über Matrizen aus nicht negativen Elementen, S.-B. Deutsch. Akad. Wiss., 23 (1912), 456–477 Google Scholar[7] F. R. Gantmacher, The theory of matrices. Vol. 2, Translated by K. A. Hirsch, Chelsea Publishing Co., New York, 1959ix+276 MR0107649 (21:6372c) Google Scholar[8] Frank Harary, A graph theoretic approach to matrix inversion by partitioning, Numer. Math., 4 (1962), 128–135 10.1007/BF01386304 MR0139545 (25:2977) CrossrefGoogle Scholar[9] B. R. Heap and , M. S. Lynn, The index of primitivity of a non-negative matrix, Numer. Math., 6 (1964), 120–141 10.1007/BF01386062 MR0163921 (29:1220) CrossrefGoogle Scholar[10] B. R. Heap and , M. S. Lynn, A graph-theoretic algorithm for the solution of a linear Diophantine problem of Frobenius, Numer. Math., 6 (1964), 346–354 10.1007/BF01386084 MR0174488 (30:4689) CrossrefGoogle Scholar[11] B. R. Heap and , M. S. Lynn, On a linear Diophantine problem of Frobenius: An improved algorithm, Numer. Math., 7 (1965), 226–231 10.1007/BF01436078 MR0176959 (31:1227) CrossrefGoogle Scholar[12] B. R. Heap and , M. S. Lynn, The structure of powers of nonnegative matrices. II. The index of maximum density, SIAM J. Appl. Math., 14 (1966), 762–777 10.1137/0114064 MR0225672 (37:1265) LinkISIGoogle Scholar[13] Jan Mařík and , Vlastimil Pták, Norms, spectra and combinatorial properties of matrices, Czechoslovak Math. J, 10 (85) (1960), 181–196 MR0120243 (22:11000) CrossrefGoogle Scholar[14] O. Ore, Theory of Graphs, Colloquium Publication 38, American Mathematical Society, Providence, 1962 CrossrefGoogle Scholar[15] Peter Perkins, A theorem on regular matrices, Pacific J. Math., 11 (1961), 1529–1533 MR0147420 (26:4936) CrossrefGoogle Scholar[16] Oskar Perron, Zur Theorie der Matrices, Math. Ann., 64 (1907), 248–263 10.1007/BF01449896 MR1511438 CrossrefGoogle Scholar[17] Vlastimil Pták, On a combinatorial theorem and its application to nonnegative matrices., Czechoslovak Math. J, 8(83) (1958), 487–495 MR0110644 (22:1520a) Google Scholar[18] Vlastimil Pták and , Irži Sedlaček, On the index of imprimitivity of nonnegative matrices, Czechoslovak Math. J, 8(83) (1958), 496–501 MR0110645 (22:1520b) Google Scholar[19] David Rosenblatt, On the graphs and asymptotic forms of finite Boolean relation matrices and stochastic matrices, Naval. Res. Logist. Quart., 4 (1957), 151–167 MR0090910 (19,889d) CrossrefGoogle Scholar[20] Richard S. Varga, Matrix iterative analysis, Prentice-Hall Inc., Englewood Cliffs, N.J., 1962xiii+322 MR0158502 (28:1725) Google Scholar[21] Helmut Wielandt, Unzerlegbare, nicht negative Matrizen, Math. Z., 52 (1950), 642–648 10.1007/BF02230720 MR0035265 (11,710g) CrossrefGoogle Scholar[22] Ju. I. Ljubic, Bounds for the optimum determinization of a nondeterministic autonomous automaton, Sibirsk Mat. Ž., 5 (1964), 337–355 Google Scholar[23] Ju. I. Ljubič, Estimates for the number of states arising from the determination of an indeterminate autonomous automaton, Dokl. Akad. Nauk SSSR, 155 (1964), 41–43 MR0160727 (28:3938) Google Scholar Previous article FiguresRelatedReferencesCited ByDetails The stable index of 0-1 matricesLinear Algebra and its Applications, Vol. 600 | 1 Sep 2020 Cross Ref On k-idempotent 0-1 matricesLinear Algebra and its Applications, Vol. 597 | 1 Jul 2020 Cross Ref Subspace controllability of Boolean networksAsian Journal of Control, Vol. 21, No. 6 | 1 May 2019 Cross Ref Words of Minimum Rank in Deterministic Finite AutomataDevelopments in Language Theory | 10 July 2019 Cross Ref Level Eulerian PosetsGraphs and Combinatorics, Vol. 29, No. 4 | 24 April 2012 Cross Ref COMPETITION INDICES OF TOURNAMENTSBulletin of the Korean Mathematical Society, Vol. 45, No. 2 | 31 May 2008 Cross Ref The exponent and circumdiameter of primitive digraphsLinear Algebra and its Applications, Vol. 396 | 1 Feb 2005 Cross Ref On the generalized indices of boolean matricesLinear and Multilinear Algebra, Vol. 49, No. 2 | 1 Oct 2001 Cross Ref An inequality for non-negative matricesLinear Algebra and its Applications, Vol. 290, No. 1-3 | 1 Mar 1999 Cross Ref On the oscillating power sequence of a fuzzy matrixFuzzy Sets and Systems, Vol. 93, No. 1 | 1 Jan 1998 Cross Ref On the indices of convergence of reducible boolean matrices with the generalized period f0 ⩾ 2Linear Algebra and its Applications, Vol. 248 | 1 Nov 1996 Cross Ref The index set problem for Boolean (or nonnegative) matricesDiscrete Mathematics, Vol. 123, No. 1-3 | 1 Dec 1993 Cross Ref On the set of indices of convergence for reducible matricesLinear Algebra and its Applications, Vol. 148 | 1 Apr 1991 Cross Ref On the index of maximum density for irreducible Boolean matricesDiscrete Applied Mathematics, Vol. 21, No. 2 | 1 Oct 1988 Cross Ref On the index of convergence of an irreducible Boolean matrixLinear Algebra and its Applications, Vol. 97 | 1 Dec 1987 Cross Ref Bounds for exponents of doubly stochastic primitive matricesMathematische Zeitschrift, Vol. 137, No. 1 | 1 Mar 1974 Cross Ref On fully indecomposable matricesJournal of Computer and System Sciences, Vol. 5, No. 6 | 1 Dec 1971 Cross Ref On exponents of primitive matricesNumerische Mathematik, Vol. 18, No. 2 | 1 Apr 1971 Cross Ref A Property of Infinite Products of Boolean MatricesN. J. PullmanSIAM Journal on Applied Mathematics, Vol. 15, No. 4 | 13 July 2006AbstractPDF (286 KB)Questions & AnswersThe American Statistician, Vol. 21, No. 3 | 1 Jun 1967 Cross Ref The Structure of Powers of Nonnegative Matrices II. The Index of Maximum DensityB. R. Heap and M. S. LynnSIAM Journal on Applied Mathematics, Vol. 14, No. 4 | 3 August 2006AbstractPDF (1267 KB) Volume 14, Issue 3| 1966SIAM Journal on Applied Mathematics417-639 History Submitted:10 May 1965Published online:28 July 2006 InformationCopyright © 1966 © Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/0114052Article page range:pp. 610-639ISSN (print):0036-1399ISSN (online):1095-712XPublisher:Society for Industrial and Applied Mathematics