Structural properties of a symmetric Toeplitz and Hankel matrices
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866915095411425280 |
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| author | Chu, Hojin Ryu, Homoon |
| author_facet | Chu, Hojin Ryu, Homoon |
| contents | In this paper, we investigate properties of a symmetric Toeplitz matrix and a Hankel matrix by studying the components of its graph. To this end, we introduce the notion of ``weighted Toeplitz graph" and ``weighted Hankel graph", which are weighted graphs whose adjacency matrix are a symmetric Toeplitz matrix and a Hankel matrix, respectively. By studying the components of a weighted Toeplitz graph, we show that the Frobenius normal form of a symmetric Toeplitz matrix is a direct sum of symmetric irreducible Toeplitz matrices. Similarly, by studying the components of a weighted Hankel matrix, we show that the Frobenius normal form of a Hankel matrix is a direct sum of irreducible Hankel matrices. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2410_13129 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Structural properties of a symmetric Toeplitz and Hankel matrices Chu, Hojin Ryu, Homoon Combinatorics 05C22, 05C50, 15B05 In this paper, we investigate properties of a symmetric Toeplitz matrix and a Hankel matrix by studying the components of its graph. To this end, we introduce the notion of ``weighted Toeplitz graph" and ``weighted Hankel graph", which are weighted graphs whose adjacency matrix are a symmetric Toeplitz matrix and a Hankel matrix, respectively. By studying the components of a weighted Toeplitz graph, we show that the Frobenius normal form of a symmetric Toeplitz matrix is a direct sum of symmetric irreducible Toeplitz matrices. Similarly, by studying the components of a weighted Hankel matrix, we show that the Frobenius normal form of a Hankel matrix is a direct sum of irreducible Hankel matrices. |
| title | Structural properties of a symmetric Toeplitz and Hankel matrices |
| topic | Combinatorics 05C22, 05C50, 15B05 |
| url | https://arxiv.org/abs/2410.13129 |