On the condition of characteristic polynomials
arXiv:1510.04419h-index: 36
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Provides a theoretical justification for a long-standing practical guideline in numerical linear algebra.
The paper proves that the expected log condition number of zeros of characteristic polynomials of complex Gaussian matrices is Ω(n), explaining why eigenvalue computation via root-finding is numerically unstable.
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 may provide an explanation for the common wisdom in numerical linear algebra that advises against computing eigenvalues via root-finding for characteristic polynomials.