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Given two vectors $\boldsymbol\alpha=\left(\alpha_1,...,\alpha_N\right)$ and $\boldsymbol\beta=\left(\beta_1,...,\beta_N\right)$, is there an easy way to compute the eigenvalues of a matrix $M_{k,q}$ whose entries are expressed as

$$ M_{k,q}=\frac{\alpha_k}{\beta_q},\quad k=1,...,N,\quad q=1,...,N\quad ? $$

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up vote 2 down vote accepted

Hint: $M$ has rank $1$, and $M {\bf \alpha}^T = \ldots$

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