A note on Johansen's rank conditions and the Jordan form of a matrix

This note presents insights on the Jordan structure of a matrix which are derived from an extension of the I(1) and I(2) conditions in Johansen (1996). It is first observed that these conditions not only characterize, as it is well known, the size (1 or 2) of the largest Jordan block in the Jordan form of the companion matrix but more generally the geometric multiplicities, the algebraic multiplicities and the whole Jordan structure for eigenvalues of index 1 or 2. In the context of the Granger representation theorem, this means that the Johansen rank conditions do more than determine the order of integration of the process. It is then shown that an extension of these conditions leads to the characterization of the Jordan structure of any matrix.