Characteristic polynomials of typical matrices are ill-conditioned

Peter Buergisser, Felipe Cucker, Elisa Rocha Cardozo

Published 2015-10-15Version 1

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 gives a rigorous justification of the common wisdom in numerical linear algebra that advises against computing eigenvalues via root-finding for characteristic polynomials.