2019
DOI: 10.48550/arxiv.1906.11819
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Gaussian Regularization of the Pseudospectrum and Davies' Conjecture

Abstract: A matrix A ∈ C n×n is diagonalizable if it has a basis of linearly independent eigenvectors. Since the set of nondiagonalizable matrices has measure zero, every A ∈ C n×n is the limit of diagonalizable matrices. We prove a quantitative version of this fact conjectured by E.B. Davies: for each δ ∈ (0, 1), every matrix A ∈ C n×n is at least δ A -close to one whose eigenvectors have condition number at worst c n /δ, for some constants c n dependent only on n. Our proof uses tools from random matrix theory to show… Show more

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Cited by 7 publications
(43 citation statements)
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“…Notably, the theorems above do not address whether a real matrix can be regularized by a real perturbation, and [BKMS19] left this as an open question, which we resolve, albeit with a worse dependence on .…”
Section: Related Workmentioning
confidence: 81%
See 4 more Smart Citations
“…Notably, the theorems above do not address whether a real matrix can be regularized by a real perturbation, and [BKMS19] left this as an open question, which we resolve, albeit with a worse dependence on .…”
Section: Related Workmentioning
confidence: 81%
“…For a more detailed argument see the proof of Lemma 3.1 in [BKMS19]. Substituting this inequality in the bounds above yields the advertised result.…”
Section: Bounds In Expectationmentioning
confidence: 86%
See 3 more Smart Citations