Wednesday, 16 December 2015

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 pi0,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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