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On the Number of Positive Entries in the Powers of a Non-Negative Matrix

Published online by Cambridge University Press:  20 November 2018

N. Pullman
N. Pullman*
Affiliation:
McGill University
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A real matrix A is said to be non-negative if and only if none of its entries is negative. Suppose A is an r by r non-negative matrix. We want to examine:

  1. (A) The first power of A to maximize the number of positive entries in An,

  2. (B) For each 1 ≤ i ≤ r the first power of A to maximize the number of positive entries in the i-th row of An.

We shall call the former first power the index of A and the latter the i-th row index of A (index (i, A)).

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 7 , Issue 4 , December 1964 , pp. 525 - 537
Copyright
Copyright © Canadian Mathematical Society 1964

References

1. Wielandt, H., Unzerlegbare nicht negative Matrizen, Math. Zeit. 52 (1950) pp. 642–648.CrossRefGoogle Scholar
2. Rosenblatt, D., Asymptotic Forms of Two Classes of Matrices, Naval Logistics Quarterly 4 (1957) pp. 151–165.CrossRefGoogle Scholar
3. Holladay, J. C. and Varga, R.S., On Powers of Non-Negative Matrices, A. M. S. Proceedings 9 (1958) pp. 631–634.CrossRefGoogle Scholar
4. Dulmage, A. L. and Mendelsohn, N.S., The Exponent of a Primitive Matrix, Can. Math. Bulletin 5 (1962) pp. 241–244.CrossRefGoogle Scholar
5. Wedderburn, J. H. M., Boolean Linear Associative Algebra, Ann. Math. 35 (1934) pp. 185–194.CrossRefGoogle Scholar
6. Mann, H. B. and Ryser, H. J., Systems of Distinct Representatives, Amer. Math. Mon. 60 (1953) pp. 397–401.CrossRefGoogle Scholar
7. Ore, O., Graphs and Matching Theorems, Duke Math. J. 22 (1955) p. 635.CrossRefGoogle Scholar
8. Ore, O., Theory of Graphs, A.M.S. Colloquium Publ. No. 38(1962).Google Scholar
9. Doob, J. L., Stochastic Processes, Wiley, N.Y. 1953.Google Scholar
10. Pták, V., On a Combinatorial Theorem and Its Application to Non-negative Matrices, Czechoslovakian Math. J., vol.8 (83), (1958) pp. 487–492.CrossRefGoogle Scholar