On Efficient Sampling Schemes for the Eigenvalues of Complex Wishart Matrices

TL;DR

This paper reviews and compares efficient sampling methods for the eigenvalues of complex Wishart matrices, highlighting existing approaches involving tridiagonal and bidiagonal matrix factorizations, applicable to both complex and real cases.

Contribution

It clarifies and consolidates two existing efficient sampling schemes for Wishart eigenvalues, extending their applicability to spiked variance matrices and real Wishart matrices.

Findings

Two distinct efficient sampling methods for Wishart eigenvalues are identified.

Methods involve eigenvalues of tridiagonal and bidiagonal matrix factorizations.

Approaches are applicable to both complex and real Wishart matrices.

Abstract

The paper "An efficient sampling scheme for the eigenvalues of dual Wishart matrices", by I.~Santamar\'ia 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.