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Input-Output Tables and Some Theory of Defective Matrices

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  • Mohit Arora
  • Deepankar Basu

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.

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  • Mohit Arora & Deepankar Basu, 2022. "Input-Output Tables and Some Theory of Defective Matrices," Papers 2208.00226, arXiv.org.
    • Handle: RePEc:arx:papers:2208.00226
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    References listed on IDEAS

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    1. Heinz Kurz, 2011. "Who is Going to Kiss Sleeping Beauty? On the 'Classical' Analytical Origins and Perspectives of Input-Output Analysis," Review of Political Economy, Taylor & Francis Journals, vol. 23(1), pages 25-47.
    2. Ernest Liu & Aleh Tsyvinski, 2020. "Dynamical Structure and Spectral Properties of Input-Output Networks," NBER Working Papers 28178, National Bureau of Economic Research, Inc.
    3. Fremdling Rainer & Staeglin Reiner, 2014. "An Input-Output Table for Germany in 1936: A Documentation of Results, Sources and Research Strategy," Jahrbuch für Wirtschaftsgeschichte / Economic History Yearbook, De Gruyter, vol. 55(2), pages 187-298, November.
    4. Vasco M. Carvalho & Alireza Tahbaz-Salehi, 2019. "Production Networks: A Primer," Annual Review of Economics, Annual Reviews, vol. 11(1), pages 635-663, August.
    5. Heinz Kurz & Neri Salvadori, 2000. "'Classical' Roots of Input-Output Analysis: A Short Account of its Long Prehistory," Economic Systems Research, Taylor & Francis Journals, vol. 12(2), pages 153-179.
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