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Distribution of Eigenvalues of Real Symmetric Palindromic Toeplitz Matrices and Circulant Matrices

Author

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  • Adam Massey

    (Brown University)

  • Steven J. Miller

    (Brown University)

  • John Sinsheimer

    (The Ohio State University)

Abstract

Consider the ensemble of real symmetric Toeplitz matrices, each independent entry an i.i.d. random variable chosen from a fixed probability distribution p of mean 0, variance 1, and finite higher moments. Previous investigations showed that the limiting spectral measure (the density of normalized eigenvalues) converges weakly and almost surely, independent of p, to a distribution which is almost the standard Gaussian. The deviations from Gaussian behavior can be interpreted as arising from obstructions to solutions of Diophantine equations. We show that these obstructions vanish if instead one considers real symmetric palindromic Toeplitz matrices, matrices where the first row is a palindrome. A similar result was previously proved for a related circulant ensemble through an analysis of the explicit formulas for eigenvalues. By Cauchy’s interlacing property and the rank inequality, this ensemble has the same limiting spectral distribution as the palindromic Toeplitz matrices; a consequence of combining the two approaches is a version of the almost sure Central Limit Theorem. Thus our analysis of these Diophantine equations provides an alternate technique for proving limiting spectral measures for certain ensembles of circulant matrices.

Suggested Citation

  • Adam Massey & Steven J. Miller & John Sinsheimer, 2007. "Distribution of Eigenvalues of Real Symmetric Palindromic Toeplitz Matrices and Circulant Matrices," Journal of Theoretical Probability, Springer, vol. 20(3), pages 637-662, September.
    • Handle: RePEc:spr:jotpro:v:20:y:2007:i:3:d:10.1007_s10959-007-0078-x
    DOI: 10.1007/s10959-007-0078-x
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    References listed on IDEAS

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    1. Bose, Arup & Mitra, Joydip, 2002. "Limiting spectral distribution of a special circulant," Statistics & Probability Letters, Elsevier, vol. 60(1), pages 111-120, November.
    2. Berkes, István & Csáki, Endre, 2001. "A universal result in almost sure central limit theory," Stochastic Processes and their Applications, Elsevier, vol. 94(1), pages 105-134, July.
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    Cited by:

    1. Arup Bose & Joydip Mitra & Arnab Sen, 2012. "Limiting Spectral Distribution of Random k-Circulants," Journal of Theoretical Probability, Springer, vol. 25(3), pages 771-797, September.
    2. Steven Jackson & Steven J. Miller & Thuy Pham, 2012. "Distribution of Eigenvalues of Highly Palindromic Toeplitz Matrices," Journal of Theoretical Probability, Springer, vol. 25(2), pages 464-495, June.
    3. Dang-Zheng Liu & Zheng-Dong Wang, 2011. "Limit Distribution of Eigenvalues for Random Hankel and Toeplitz Band Matrices," Journal of Theoretical Probability, Springer, vol. 24(4), pages 988-1001, December.
    4. Murat Koloğlu & Gene S. Kopp & Steven J. Miller, 2013. "The Limiting Spectral Measure for Ensembles of Symmetric Block Circulant Matrices," Journal of Theoretical Probability, Springer, vol. 26(4), pages 1020-1060, December.

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