A family of doubly stochastic matrices involving Chebyshev polynomials
A doubly stochastic matrix is a square matrix \(A=(a_{ij})\) of non-negative real numbers such that \(\sum_{i}a_{ij}=\sum_{j}a_{ij}=1\). The Chebyshev polynomial of the first kind is defined by the recurrence relation \(T_0(x)=1, T_1(x)=x\), and \[T_{n+1}(x)=2xT_n(x)-T_{n-1}(x).\]In this paper, we s...
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| Date: | 2019 |
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| Main Authors: | , |
| Format: | Article |
| Language: | English |
| Published: |
Lugansk National Taras Shevchenko University
2019
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| Subjects: | |
| Online Access: | https://admjournal.luguniv.edu.ua/index.php/adm/article/view/557 |
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| Journal Title: | Algebra and Discrete Mathematics |
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Algebra and Discrete Mathematics| Summary: | A doubly stochastic matrix is a square matrix \(A=(a_{ij})\) of non-negative real numbers such that \(\sum_{i}a_{ij}=\sum_{j}a_{ij}=1\). The Chebyshev polynomial of the first kind is defined by the recurrence relation \(T_0(x)=1, T_1(x)=x\), and \[T_{n+1}(x)=2xT_n(x)-T_{n-1}(x).\]In this paper, we show a \(2^k\times 2^k\) (for each integer \(k\geq 1\)) doubly stochastic matrix whose characteristic polynomial is \(x^2-1\) times a product of irreducible Chebyshev polynomials of the first kind (up to rescaling by rational numbers). |
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