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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2407.21680 |
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| _version_ | 1866917402162233344 |
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| author | Casper, W. Riley Zurrian, Ignacio |
| author_facet | Casper, W. Riley Zurrian, Ignacio |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_21680 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The Pascal Matrix, Commuting Tridiagonal Operators and Fourier Algebras Casper, W. Riley Zurrian, Ignacio Spectral Theory Numerical Analysis 7K35, 16S32, 39A70 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. |
| title | The Pascal Matrix, Commuting Tridiagonal Operators and Fourier Algebras |
| topic | Spectral Theory Numerical Analysis 7K35, 16S32, 39A70 |
| url | https://arxiv.org/abs/2407.21680 |