## Operators associated with a pair of nonnegative matrices

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- by Gerald E. Suchan
- Proc. Amer. Math. Soc.
**35**(1972), 53-54 - DOI: https://doi.org/10.1090/S0002-9939-1972-0296092-6
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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.

## References

- 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**206019**, DOI 10.1016/0022-247X(66)90184-3 - 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**225802**, DOI 10.1090/S0002-9947-1968-0225802-2

## Bibliographic Information

- © Copyright 1972 American Mathematical Society
- 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