A limiting property for the powers of a non-negative, reducible matrix

Considering a dynamic multisector model, the behaviour of A^k is examined for a non-negative matrix A as k becomes large. It is well known that for a primitive matrix A, the matrix ( A λ)^k converges to yq' (q'y), where λ denotes the dominant eigenvalue, and where y and q' are the right and left eigenvector associated with λ. In this note, the same result is shown to hold, under certain conditions, when A is reducible with primitive diagonal block submatrices. Under weaker conditions A^k (e′A^ke) is proved to converge to yq′ (e′y)(q′e), where e denotes the summation vector. The results are interpreted in terms of dynamic multisector models and interindustry linkage indicators.