  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 propagation relies on eigendecomposition of
relevant input-output matrices. It is well known that only diagonalizable
matrices can be eigendecomposed; 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
eigenvalues. To apply our results, we offer examples of input-output tables
from India in the 1950s that were not diagonalizable and were hence, defective.
