Distribution of Eigenvalues of Real Symmetric Palindromic Toeplitz Matrices and Circulant Matrices
By Adam Massey, Steven J. Miller, J. Sinsheimer. 2005
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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.
Subfields: Applied Mathematics, Computational Theory and Mathematics, Statistics and Probability
View Paper: https://www.semanticscholar.org/paper/a1f24171d0d44db5731040454f15d11e491ceeb0
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In Applied Mathematics
In Computational Theory and Mathematics
In Statistics and Probability
Citations in the corpus
- Another look at the moment method for large dimensional random matrices (2008, methodology): “This imposes other restrictions on the entries and we obtain the following palindromic Toeplitz matrix, see Massey, Miller and Sinsheimer [15], 2006) [MMS]:”