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Integration of invariant matrices and moments of inverses of Ginibre and Wishart matrices

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  • Collins, Benoît
  • Matsumoto, Sho
  • Saad, Nadia

Abstract

We consider random matrices that have invariance properties under the action of unitary groups (either a left–right invariance, or a conjugacy invariance), and we give formulas for moments in terms of functions of eigenvalues. Our main tool is the Weingarten calculus. As an application, we obtain new formulas for the pseudoinverse of Gaussian matrices and for the inverse of compound Wishart matrices.

Suggested Citation

  • Collins, Benoît & Matsumoto, Sho & Saad, Nadia, 2014. "Integration of invariant matrices and moments of inverses of Ginibre and Wishart matrices," Journal of Multivariate Analysis, Elsevier, vol. 126(C), pages 1-13.
    • Handle: RePEc:eee:jmvana:v:126:y:2014:i:c:p:1-13
    DOI: 10.1016/j.jmva.2013.12.011
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    References listed on IDEAS

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    1. Baik, Jinho & Silverstein, Jack W., 2006. "Eigenvalues of large sample covariance matrices of spiked population models," Journal of Multivariate Analysis, Elsevier, vol. 97(6), pages 1382-1408, July.
    2. Carlos M. Carvalho & Hélène Massam & Mike West, 2007. "Simulation of hyper-inverse Wishart distributions in graphical models," Biometrika, Biometrika Trust, vol. 94(3), pages 647-659.
    3. Zdzisław Burda & Andrzej Jarosz & Maciej Nowak & Jerzy Jurkiewicz & Gabor Papp & Ismail Zahed, 2011. "Applying free random variables to random matrix analysis of financial data. Part I: The Gaussian case," Quantitative Finance, Taylor & Francis Journals, vol. 11(7), pages 1103-1124.
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