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Maximal eigenvalue of a real symmetric Toeplitz matrix

The $n×n$ matrix $A_n$ is defined by the elements $a_{ij}=n−|i−j|$. \begin{bmatrix} n & n-1 & n-2 & \cdots & 1\\ n-1 & n & n-1 & \cdots & 2\\ n-2 & n-1 & n & \cdots & 3\\ \vdots & \vdots & \vdots & & …
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