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

Abstract

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 k e) 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.