The following matrix shows up in studying multinomial distributions as the covariance matrix. Let p be a column vector of dimension k with pi≥0,∑ki=1pi=1. Let
A:=Diag(p)−ppT,
where 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
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