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arXiv · 2311.06810

On the trace-zero doubly stochastic matrices of order 5

Amrita Mandal·Bibhas Adhikari
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Abstract

We propose a graph theoretic approach to determine trace of product of two permutation matrices through a weighted digraph representation of the permutation matrices. Consequently, we derive trace-zero doubly stochastic (DS) matrices of order $5$ whose $k$-th power is also a trace-zero DS matrix for $k\in\{2,3,4,5\}$. Then, we determine necessary conditions for the coefficients of a generic polynomial of degree $5$ to be realizable as the characteristic polynomial of a trace-zero DS matrix of order $5$. Finally, we approximate the eigenvalue region of trace-zero DS matrices of order $5.$

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Amrita Mandal, Bibhas Adhikari. 2024-10-20. On the trace-zero doubly stochastic matrices of order 5. https://arxiv.org/abs/2311.06810

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