linear algebra - Eigenvalues of a multinomial covariance matrix

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?

Answer

In general no closed form. See this paper:

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