Using some concepts and tools from Functional Analysis, we express explicit forms of ‘eigenvectors’ of two Toeplitz operators: An operator of multiplicity one, which appears e.g. in Autoregressive Process, and a banded operator of high multiplicity. Then we investigate the approximate eigenvectors of their associated Toeplitz matrices of large sizes, and after some numerical experiments, the paper concludes with an application in the principal component analysis method for dimension reduction.

中文翻译：

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一些托普利茨算子和矩阵的近似对角化

使用泛函分析中的一些概念和工具，我们表达了两个 Toeplitz 算子的“特征向量”的显式形式：一个是多重性算子，出现在例如自回归过程中，另一个是高多重性的带状算子。然后我们研究了它们相关联的大尺寸Toeplitz矩阵的近似特征向量，经过一些数值实验，论文总结了主成分分析方法在降维方面的应用。