The Pascal Matrix, Commuting Tridiagonal Operators and Fourier Algebras
This work provides new theoretical insights into the structure of the Pascal matrix and its commuting operators, which may benefit numerical linear algebra and related fields, but the results are incremental.
The authors prove the existence of symmetric tridiagonal matrices commuting with the Pascal matrix, provide explicit expressions, and show that these matrices enable numerically stable diagonalization of the Pascal matrix.
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.