Operators associated with a pair of nonnegative matrices

Let A m × n , B m × n , X n × 1 {A_{m \times n}},{B_{m \times n}},{X_{n \times 1}} , and Y m × 1 {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 ( U X ) i = X i − 1 [ or ( U Y ) i = Y i − 1 ] {(UX)_i} = X_i^{ - 1}[{\text {or}}{(UY)_i} = Y_i^{ - 1}] , T = U B t U A T = U{B^t}UA and T ′ = U A U B t T’ = UAU{B^t} . T is called irreducible if for no nonempty proper subset S of { 1 , ⋯ , n } \{ 1, \cdots ,n\} it is true that X i = 0 , i ∈ S ; X i ≠ 0 , i ∉ S {X_i} = 0,i \in S;{X_i} \ne 0,i \notin S , implies ( T X ) i = 0 , i ∈ S ; ( T X ) i ≠ 0 , i ∉ S {(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 {A_1} and A 2 {A_2} , a positive number θ \theta , and two diagonal matrices D and E with positive main diagonal entries such that D A E = A 1 DAE = {A_1} and θ D B E = A 2 t \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.