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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Operators associated with a pair of nonnegative matrices


Author: Gerald E. Suchan
Journal: Proc. Amer. Math. Soc. 35 (1972), 53-54
MSC: Primary 15A48
DOI: https://doi.org/10.1090/S0002-9939-1972-0296092-6
MathSciNet review: 0296092
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Abstract: Let $ {A_{m \times n}},{B_{m \times n}},{X_{n \times 1}}$, and $ {Y_{m \times 1}}$ be matrices whose entries are nonnegative real numbers and suppose that no row of A and no column of B consists entirely of zeroes. Define the operators U, T and T' by $ {(UX)_i} = X_i^{ - 1}[{\text{or}}{(UY)_i} = Y_i^{ - 1}]$, $ T = U{B^t}UA$ and $ T' = UAU{B^t}$. T is called irreducible if for no nonempty proper subset S of $ \{ 1, \cdots ,n\} $ it is true that $ {X_i} = 0,i \in S;{X_i} \ne 0,i \notin S$, implies $ {(TX)_i} = 0,i \in S;{(TX)_i} \ne 0,i \notin S$. M. V. Menon proved the following Theorem. If T is irreducible, there exist row-stochastic matrices $ {A_1}$ and $ {A_2}$, a positive number $ \theta $, and two diagonal matrices D and E with positive main diagonal entries such that $ DAE = {A_1}$ and $ \theta DBE = A_2^t$. Since an analogous theorem holds for T', it is natural to ask if it is possible that T' be irreducible if T is not. It is the intent of this paper to show that T' is irreducible if and only if T is irreducible.


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DOI: https://doi.org/10.1090/S0002-9939-1972-0296092-6
Keywords: Nonnegative matrix, stochastic, irreducible
Article copyright: © Copyright 1972 American Mathematical Society