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The First Two Eigenvalues of Large Random Matrices and Brody's Hypothesis on the Stability of Large Input-Output Systems

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  • Guang-Zhen Sun

Abstract

Brody (1997) notices that for large random Leontief matrices, namely non-negative square matrices with all entries i.i.d., the ratio between the subdominant eigenvalue (in modulus) and the dominant eigenvalue declines generically to zero at a speed of the square root of the size of the matrix as the matrix size goes to infinity. Since then, several studies have been published in this journal in attempting to rigorously verify Brody's conjecture. This short article, drawing upon some theorems obtained in recent years in the literature on empirical spectral distribution of random matrices, offers a short proof of Brody's conjecture, and discusses briefly some related issues.

Suggested Citation

  • Guang-Zhen Sun, 2008. "The First Two Eigenvalues of Large Random Matrices and Brody's Hypothesis on the Stability of Large Input-Output Systems," Economic Systems Research, Taylor & Francis Journals, vol. 20(4), pages 429-432.
    • Handle: RePEc:taf:ecsysr:v:20:y:2008:i:4:p:429-432
    DOI: 10.1080/09535310802551471
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    Citations

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    Cited by:

    1. Theodore Mariolis & Lefteris Tsoulfidis, 2012. "On Brody’S Conjecture: Facts And Figures From The Us Economy," Discussion Paper Series 2012_06, Department of Economics, University of Macedonia, revised May 2012.
    2. Luis Daniel Torres-González, 2020. "The Characteristics of the Productive Structure Behind the Empirical Regularities in Production Prices Curves," Working Papers 2016, New School for Social Research, Department of Economics.
    3. Anwar Shaikh & Luiza Nassif, 2018. "Eigenvalue distribution, matrix size and the linearity of wage-profit curves," Working Papers 1812, New School for Social Research, Department of Economics.
    4. Torres-González, Luis Daniel, 2022. "The Characteristics of the Productive Structure Behind the Empirical Regularities in Production Prices Curves," Structural Change and Economic Dynamics, Elsevier, vol. 62(C), pages 622-659.
    5. Iliadi, Fotoula & Mariolis, Theodore & Soklis, George & Tsoulfidis, Lefteris, 2012. "Bienenfeld’s approximation of production prices and eigenvalue distribution: some more evidence from five European economies," MPRA Paper 36282, University Library of Munich, Germany.
    6. Mariolis, Theodore & Tsoulfidis, Lefteris, 2010. "Eigenvalue distribution and the production price-profit rate relationship in linear single-product systems: theory and empirical evidence," MPRA Paper 43716, University Library of Munich, Germany.

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