Condition Numbers of Indefinite Rank 2 Ghost Wishart Matrices

TL;DR

This paper derives the eigenvalue ratio distribution and condition number formulas for indefinite Wishart matrices of size 2, applicable across various algebraic settings, with validation through numerical experiments.

Contribution

It provides explicit formulas for the eigenvalue ratio and condition number distributions of indefinite Wishart matrices of size 2 for different algebraic cases, including ghost eta.

Findings

Derived eigenvalue ratio distribution for indefinite Wishart matrices.

Formulas for condition number distribution across real, complex, quaternionic, and ghost cases.

Validated theoretical results with numerical experiments.

Abstract

We define an indefinite Wishart matrix as a matrix of the form A=W^{T}W\Sigma, where \Sigma is an indefinite diagonal matrix and W is a matrix of independent standard normals. We focus on the case where W is L by 2 which has engineering applications. We obtain the distribution of the ratio of the eigenvalues of A. This distribution can be "folded" to give the distribution of the condition number. We calculate formulas for W real (\beta=1), complex (\beta=2), quaternionic (\beta=4) or any ghost 0<\beta<\infty. We then corroborate our work by comparing them against numerical experiments.