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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?

Hedonist
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  • Maybe relevant: http://math.stackexchange.com/questions/730134/determinant-of-rank-one-perturbation-of-a-diagonal-matrix http://math.stackexchange.com/questions/506761/maximum-eigenvalue-of-a-diagonal-plus-rank-one-matrix – Jean-Claude Arbaut Oct 08 '15 at 19:36

1 Answers1

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In general no closed form. See this paper:

https://projecteuclid.org/download/pdfview_1/euclid.bjps/1405603508

amWhy
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Mingyue Gao
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  • 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). – Lee David Chung Lin Nov 16 '18 at 02:54