Characteristic polynomials of typical matrices are ill-conditioned. Bürgisser, P., Cucker, F., & Cardozo, E. R. 10 2015. Paper abstract bibtex 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 Ω(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.

@UNPUBLISHED{BCC-Characteristic-Polynomials-Of-Typical-Matrices-Are-Ill-conditioned,
TITLE={Characteristic polynomials of typical matrices are ill-conditioned},
URL={http://arxiv.org/abs/1510.04419},
AUTHOR={Peter Bürgisser and Felipe Cucker and Elisa Rocha Cardozo},
YEAR={2015},
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 Ω(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.},
MONTH={10}
}

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