实对称回文Toeplitz矩阵和循环矩阵的特征值分布
Distribution of Eigenvalues of Real Symmetric Palindromic Toeplitz Matrices and Circulant Matrices
摘要：
考虑实对称Toeplitz矩阵的系综，每个独立元素是独立同分布的。从均值为0、方差为1和有限高阶矩的固定概率分布p中选择的随机变量。以前的研究表明，限制谱测度（归一化本征值的密度）收敛弱，几乎肯定，独立于p，几乎是标准的高斯分布。从高斯行为的偏差可以解释为所产生的障碍，以解决丢番图方程。我们表明，这些障碍消失，而不是考虑真正的对称回文Toeplitz矩阵，矩阵的第一行是回文。类似的结果以前证明了相关的循环系综通过分析显式公式的特征值。通过柯西的交错性质和秩不等式，这个集合具有与回文Toeplitz矩阵相同的极限谱分布;结合这两种方法的结果是几乎必然的中心极限定理。因此，我们的分析这些丢番图方程提供了一种替代技术，证明限制谱测度的某些合奏循环矩阵。
英文摘要：
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.
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