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Partially decomposable and totally indecomposable nonnegative matrices

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Abstract

We considerm×n (m≤n) matrices with entries from an arbitrary given finite set of nonnegative real numbers, including zero. In particular, (0, 1)-matrices are studied. On the basis of the classification of such matrices by type and of the general formula for the number of matrices of nullityt valid fort>n andt≥n>m (see [2]), an asymptotic (asn → ∞) expansion is obtained for the total number of: (a) totally indecomposable matrices (Theorems 1 and 5), (b) partially decomposable matrices of given nullityt≥n (Theorems 2 and 4), (c) matrices with zero permanent (without using the inclusion-exclusion principle; Corollary of Theorem 2).

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References

  1. T. I. Fennet and G. Loizou, “Combinatorial aspects of rectangular non-negative matrices,”Discrete Math.,20, 217–234 (1977).

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  2. Yu. V. Bolotnikov and V. E. Tarakanov, “Nonnegative matrices with zero permanent,”Mat. Zametki [Math. Notes],58, No. 4, 493–504 (1995).

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  3. V. N. Sachkov,Probabilistic Methods in Combinatorial Analysis [in Russian], Nauka, Moscow (1978).

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Authors and Affiliations

  1. Steklov Mathematics Institute, Russian Academy of Sciences, USSR

    Y. V. Bolotnikov & V. E. Tarakanov

Authors
  1. Y. V. Bolotnikov
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  2. V. E. Tarakanov
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Translated fromMatematicheskie Zametki, Vol. 59, No. 5, pp. 643–662, May, 1996.

The work of the second author was supported by the Russian Foundation for Basic Research under grant No. 93-011-1443.

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Bolotnikov, Y.V., Tarakanov, V.E. Partially decomposable and totally indecomposable nonnegative matrices. Math Notes 59, 463–476 (1996). https://doi.org/10.1007/BF02308812

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  • DOI: https://doi.org/10.1007/BF02308812

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