On the condition of characteristic polynomials

TL;DR

This paper demonstrates that the expected logarithm of the condition number of zeros of characteristic polynomials of Gaussian matrices grows linearly with matrix size, explaining why root-finding for eigenvalues is numerically unstable.

Contribution

It provides a theoretical lower bound on the condition number of characteristic polynomial zeros for Gaussian matrices, illuminating numerical stability issues.

Findings

Expected log condition number grows linearly with matrix size

Root-finding for eigenvalues is numerically unstable due to high condition numbers

Supports practical advice against eigenvalue computation via characteristic polynomials

Abstract

We prove that the expectation of the logarithm of the condition number of each of the zeros of the characteristic polynomial of a complex standard Gaussian matrix is ${\Omega}(n)$. This may provide an explanation for the common wisdom in numerical linear algebra that advises against computing eigenvalues via root-finding for characteristic polynomials.