Ivan Shishkin, Rye (1878)

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Eigenvalues of a weird matrix

by Ancient Tree·
25
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What are the eigenvalues of the following matrix, defined for n≥3n\geq 3:
An=(cos⁡(log⁡(n))cos⁡(log⁡(n2))⋯cos⁡(log⁡(nn))sin⁡(log⁡(n))sin⁡(log⁡(n2))⋯sin⁡(log⁡(nn)))A_n=\left(\begin{array}{llll} \cos (\log (n)) & \cos \left(\log \left(n^2\right)\right) & \cdots & \cos \left(\log \left(n^n\right)\right) \\ \sin (\log (n)) & \sin \left(\log \left(n^2\right)\right) & \cdots & \sin \left(\log \left(n^n\right)\right) \end{array}\right)

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Solution by Ancient Tree

Discussions1 useful vote

This is a trick question. The matrix is not square, so the notion of eigenvalue is ill-defined.

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