On Efficient Sampling Schemes for the Eigenvalues of Complex Wishart Matrices

TL;DR

It is pointed out that existing results in the literature give two distinct ways to efficiently sample the eigenvalues in the general $n$ case and a formula involving just two classical distributions specifying the two eigenvalues in the case of n=2 is given.

Abstract

The paper "An efficient sampling scheme for the eigenvalues of dual Wishart matrices", by I.~Santamaría and V.~Elvira, [\emph{IEEE Signal Processing Letters}, vol.~28, pp.~2177--2181, 2021] \cite{SE21}, poses the question of efficient sampling from the eigenvalue probability density function of the $n \times n$ central complex Wishart matrices with variance matrix equal to the identity. Underlying such complex Wishart matrices is a rectangular $R \times n$ $(R \ge n)$ standard complex Gaussian matrix, requiring then $2Rn$ real random variables for their generation. The main result of \cite{SE21} gives a formula involving just two classical distributions specifying the two eigenvalues in the case $n=2$. The purpose of this Letter is to point out that existing results in the literature give two distinct ways to efficiently sample the eigenvalues in the general $n$ case. One is in terms of the eigenvalues of a tridiagonal matrix which factors as the product of a bidiagonal matrix and its transpose, with the $2n+1$ nonzero entries of the latter given by (the square root of) certain chi-squared random variables. The other is as the generalised eigenvalues for a pair of bidiagonal matrices, also containing a total of $2n+1$ chi-squared random variables. Moreover, these characterisation persist in the case of that the variance matrix consists of a single spike, and for the case of real Wishart matrices.

Figures (1)

• Figure 1: Empirical PDFs generated by sampling $2 \times 10^4$ times the smallest and largest eigenvalue respectively from the random tridiagonal matrix $\mathbf T_\beta$ with parameters $n=R=30$ and $\beta = 2$, compared against theoretical predictions.