The following matrix shows up in studying multinomial distributions as the covariance matrix. Let $p$ be a column vector of dimension $k$ with $p_i\geq 0, \sum_{i=1}^k p_i = 1.$ Let

$$A:=\mathrm{Diag}(p) - pp^T,$$

where $\mathrm{Diag}(p)$ is a diagonal matrix with $p$ on the diagonal.

$A$ is a positive semidefinite matrix. One of its eigenvalues is zero (corresponding to the all-ones eigenvector). What are its other eigenvalues as a function of $p$? Is there a closed-form expression for those eigenvalues?


In general no closed form. See this paper:


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  • $\begingroup$ Link-only answer (without at least some elaboration) is generally unacceptable. Either you should add some substantial descriptions, or you can wait till you have enough reputation to leave a link-only comment (which is perfectly fine). $\endgroup$ – Lee David Chung Lin Nov 16 '18 at 2:54

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