Since the Frobenius/Perron theorem was introduced nonnegative matrices can be divided in two classes. Those matrices that are reducible and those that are not. Nonnegative irreducible matrices have nice properties. The algebraic, and therefore the geometric, multiplicity of the dominant eigenvalue is always 1. This implies that a nonnegative irreducible matrix has always a simple dominant eigenvalue and a unique left-hand and right-hand Perron eigenvector. Moreover these eigenvectors are positive, which makes it in general easier to calculate them.

Nonnegative reducible matrices are causing more problems. The left-hand and right-hand Perron eigenvectors of nonnegative reducible matrices contain zeros [Gantmacher] in several, but not all, places. We call these eigenvectors semi positive. The algebraic multiplicity of the dominant eigenvalue can be greater than 1. This implies that the geometric multiplicity of the Perron eigenvectors, and therefore the dimension of the nonnegative eigenspace, is no longer certain. Our aim is to find a method that discovers the semi positive Perron eigenvectors and makes it possible to assert the dimension of the nonnegative eigenspace.

Why are we doing this? What is our interest in finding such a method? The answer to these questions will be mainly given from the econometric point of view. In closed input/output models, with multiple sectors, reducible matrices sometimes occur. The Perron eigenvectors can be seen as balanced growth paths. The right-hand Perron eigenvectors determine the balanced growth paths of the quantity produced. The positive entries in these eigenvectors show what sectors play a key role in long run production. The left-hand Perron eigenvectors show the balanced growth paths of the prices, and can be interpreted in an analogous way.

Perron eigenvectors can also be used to measure inter-industry linkages. The right-hand/ left-hand Perron eigenvectors can be used to measure forward/backward linkages respectively. This method can compete with the methods developed by Chenery-Watanabe and Rasmussen.

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