Operators associated with a pair of nonnegative matrices

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.