# Computation of the Expected Euler Characteristic for the Largest Eigenvalue of a Real Non-central Wishart Matrix

## TL;DR

This paper develops an approximate formula for the distribution of the largest eigenvalue of real Wishart matrices using the expected Euler characteristic method, including differential equations and numerical analysis for small dimensions.

## Contribution

It introduces a new approximation method for eigenvalue distributions of Wishart matrices applicable to general dimensions, with detailed analysis for 2x2 cases.

## Key findings

- Derived an approximate distribution formula for the largest eigenvalue.
- Established a differential equation for the 2x2 case.
- Performed numerical analysis validating the approximation.

## Abstract

We give an approximate formula for the distribution of the largest eigenvalue of real Wishart matrices by the expected Euler characteristic method for the general dimension. The formula is expressed in terms of a definite integral with parameters. We derive a differential equation satisfied by the integral for the $2 \times 2$ matrix case and perform a numerical analysis of it.