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 , and 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 , and . *T* is called irreducible if for no nonempty proper subset *S* of it is true that , implies . M. V. Menon proved the following Theorem. If *T* is irreducible, there exist row-stochastic matrices and , a positive number , and two diagonal matrices *D* and *E* with positive main diagonal entries such that and . 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.

**[1]**Richard A. Brualdi, Seymour V. Parter, and Hans Schneider,*The diagonal equivalence of a nonnegative matrix to a stochastic matrix*, J. Math. Anal. Appl.**16**(1966), 31–50. MR**0206019**, https://doi.org/10.1016/0022-247X(66)90184-3**[2]**M. V. Menon,*Some spectral properties of an operator associated with a pair of nonnegative matrices*, Trans. Amer. Math. Soc.**132**(1968), 369–375. MR**0225802**, https://doi.org/10.1090/S0002-9947-1968-0225802-2

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1972-0296092-6

Keywords:
Nonnegative matrix,
stochastic,
irreducible

Article copyright:
© Copyright 1972
American Mathematical Society