A note on Johansen's rank conditions and the Jordan form of a matrix
本文从Johansen (1996)的秩条件出发，揭示了矩阵Jordan结构的更多信息，不仅限于最大Jordan块的大小，还涉及特征值的几何重数、代数重数及整个Jordan结构，对理解Granger表示定理中积分阶数的确定有重要意义。
This note presents insights on the Jordan structure of a matrix which are derived from an extension of the and 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.