帕斯卡矩阵、交换三对角算子与傅里叶代数
The Pascal Matrix, Commuting Tridiagonal Operators and Fourier Algebras
AI总结：
研究帕斯卡矩阵的有限和无限版本，证明存在与之交换的对称三对角矩阵，通过分析相关傅里叶代数，展示帕斯卡矩阵的线性关系源于三个基本关系，并证明三对角矩阵的特征向量对定义二项变换的自然基，以及交换三对角矩阵可稳定地对角化帕斯卡矩阵。
摘要：
我们研究对称的帕斯卡矩阵，包括其有限和无限版本，并通过给出这些交换矩阵的显式表达式来证明其存在。这是通过研究相关的傅里叶代数实现的，作为副产品，这使我们能够证明帕斯卡矩阵条目的所有线性关系都源于三个基本关系。我们还展示三对角矩阵的特征向量对定义二项变换的自然基，最后证明交换的三对角矩阵为帕斯卡矩阵提供了数值稳定的对角化方法。
英文摘要：
We consider the (symmetric) Pascal matrix, in its finite and infinite versions, and prove the existence of symmetric tridiagonal matrices commuting with it by giving explicit expressions for these commuting matrices. This is achieved by studying the associated Fourier algebra, which as a byproduct, allows us to show that all the linear relations of a certain general form for the entries of the Pascal matrix arise from only three basic relations. We also show that pairs of eigenvectors of the tridiagonal matrix define a natural eigenbasis for the binomial transform. Lastly, we show that the commuting tridiagonal matrices provide a numerically stable means of diagonalizing the Pascal matrix.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
