Input-Output Tables and Some Theory of Defective Matrices

TL;DR

This paper explores the mathematical properties of input-output matrices in production networks, focusing on conditions for diagonalizability and analyzing historical Indian input-output tables that are defective matrices.

Contribution

It provides necessary and sufficient conditions for matrix diagonalizability based on rank and eigenvalues, with applications to real-world input-output data.

Findings

Identifies conditions for matrix diagonalizability using rank and eigenvalues.

Shows some historical Indian input-output tables are defective matrices.

Highlights implications for shock propagation analysis in production networks.

Abstract

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.