Operators Associated with a Pair of Nonnegative Matrices

Let Amx„, Bmxn, Xnxl, and Ymxl 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)t-X? [or (UY),= Y;1], T=UB'UA and T' = UAUB'. Tis called irreducible if for no nonempty proper subset S of (1, • ■ ■ , n} it is true that X,=0, ieS; X,^0, i$ S, implies (TX),=0, ieS; (TX)i^O, i $ S. M. V. Menon proved the following Theorem. If T is irreducible, there exist row-stochastic matrices Ax and At, a positive number 6, and two diagonal matrices D and E with positive main diagonal entries such that DAE=AX and 6DBE=A'2. Since an analogous theorem holds for T', it is natural to ask if it is possible that 7" be irreducible if 7" is not. It is the intent of this paper to show that 7" is irreducible if and only if 7" is irreducible. Suppose that each of m and « is a positive integer. Let AmXn and BmXn be matrices whose entries are nonnegative real numbers and suppose that no row of A and no column of B consists entirely of zeroes. Let XnXX and YmX1 be matrices whose entries are taken from the extended real nonnegative numbers. Define the operator U by (UX)i=X{'i [or (UY)¡ = YT1] and let 0"1 = 00, co_1=0, 00+00 = 00, 000 =0, and if a>0, a 00 = 00 [1]. Define the operators T and T by T=UB'UA and T' = UAUBl where Bl is the transpose of B. Clearly . m n 1 —1 (770,= (Zbj'Za,kXk ris called irreducible if for no nonempty proper subset 5 of ^={1, •••,«} is it true that X¿=0, i e S; X^O, i $ S, implies (77Q,=0, i e S; (TX)^O, i $ S. T' is defined to be irreducible analogously. M. V. Menon [2] proved the following. Theorem 1. If T is irreducible, then there exist row-stochastic matrices Ax and A2, a positive number 0, and two diagonal matrices D and E with positive main diagonal entries such that DAE=AX and 6DBE=A2. Received by the editors March 5, 1971. AMS 1970 subject classifications. Primary 15A48; Secondary 15A15, 15A51.