Input-output tables and some theory of defective matrices

Recent developments in the theory of production networks offer interesting applications and revival of input-output analysis. Some recent papers have studied the propagation of a temporary, negative shock through an input-output network. Such analyses of shock propa- gation relies on eigendecomposition of relevant input-output matrices. It is well known that only diagonalizable matrices can be eigende- composed; those that are not diagonalizable, are known as defective matrices. In this paper, we provide necessary and sufficient conditions for diagonalizability of any square matrix using its rank and eigenval- ues. To apply our results, we offer examples of input-output tables from India in the 1950s that were not diagonalizable and were hence, defective.