Re-parameterizing and reducing families of normal operators

Grandjean, Vincent

doi:10.1007/s11856-019-1834-1

Cited by 2 publications

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References 14 publications

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“…If F A is principal then the eigenspaces extend to D A . The construction of the ideal sheaf F A is quite involved, we refer the reader to [8] for details.…”

Section: Introductionmentioning

confidence: 99%

Multiparameter perturbation theory of matrices and linear operators

Parusiński

Rond²

2020

Trans. Amer. Math. Soc.

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We show that a normal matrix A with coefficient in C[[X]], X = (X1, . . . , Xn), can be diagonalized, provided the discriminant ∆A of its characteristic polynomial is a monomial times a unit. The proof is an adaptation of the algorithm of proof of Abhyankar-Jung Theorem. As a corollary we obtain the singular value decomposition for an arbitrary matrix A with coefficient in C[[X]] under a similar assumption on ∆AA * and ∆A * A.We also show real versions of these results, i.e. for coefficients in R[[X]], and deduce several results on multiparameter perturbation theory for normal matrices with real analytic, quasi-analytic, or Nash coefficients.

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“…If F A is principal then the eigenspaces extend to D A . The construction of the ideal sheaf F A is quite involved, we refer the reader to [8] for details.…”

Section: Introductionmentioning

confidence: 99%

Multiparameter perturbation theory of matrices and linear operators

Parusiński

Rond²

2020

Trans. Amer. Math. Soc.

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Monomialization of singular metrics on real surfaces

Grandjean

2019

RACSAM

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Let B be a real analytic vector bundle of rank 2 over a smooth real analytic surface S, equipped with a real analytic fiber-metric g and such that there exists a real analytic mapping of vector bundles T S → B inducing an isomorphism outside a proper sub-variety of S. Let κ be a real analytic 2-symmetric tensor field on B. Our main result, Theorem 9.2, roughly states the following: There exists a locally finite composition of points blowings-up σ : S → S such that there exists a unique pair of real analytic singular foliations F1 and F2 on Sonly with simple singularities adapted to the exceptional divisor E and orthogonal for the (regular extension of the) pull back on S of the fiber-metric g (only semi-positive definite along E) -locally simultaneously diagonalizing the pull-back on S of the original 2-symmetric tensor field κ.When S is the resolved surface of an embedded resolution of singularities of an embedded real analytic surface singularity S0 our result thus yields a local presentation of the extension h of the pull-back on S of the inner-metric of S0 at any point of the exceptional divisor E. We furthermore recover that the pulled-back inner metric h is locally quasi-isometric to the sum of the (symmetric tensor) square of differentials of (independent) monomials in the exceptional divisor E, namely, Hsiang & Pati property is satisfied at every point of the resolved surface S.CONTENTS 10.4. Local normal form for the induced metric 32 References 34such that there exists, up to permutation, a pair of O X 2 -invertible sub-modules Θ 1 and Θ 2 of Ω 1 X 2 , such that i) Each foliation Θ i admits only simple singularities adapted to E 2 . ii) Every point a 2 of X 2 admits an open neighborhood U 2 ∋ a 2 such that if σ 2 (a 2 ) is not a singular point of X 0 , then (Dσ 2 (a 2 ))(ker Θ 1 (a 2 )) and (Dσ 2 (a 2 ))(ker Θ 2 (a 2 )) are orthogonal lines of T σ 2 (a 2 ) X 0 ;iii) The pull-back of the metric g 0 | X 0 by the resolution mapping σ 2 extends on X 2 as a real analytic semi-Riemannian metric g 2 , which writes on U 2 asOf course what is also important is point iv) of the main theorem, since we unexpectedly (that is without further blowings-up) obtain the following (Corollary 10.9 and Corollary 10.12):Corollary. Under the hypotheses of Theorem 1.2 each point a 2 in E 2 admits Hsiang & Pati coordinates, namely there exists local analytic coordinates (u, v) centered at a 2 such that i) If a 2 is a smooth point of E 2 , there exists local coordinates (u, v) at a 2 such that (E 2 , a 2 ) = {u = 0}, then the (extension of the) pulled-back metric g 2 is quasi-isometric (nearby a 2 ) to the metric given byfor non-negative integer numbers l ≥ k.ii) If a 2 is a corner point of E 2 , there exists local coordinates (u, v) at a 2 such that (E 2 , a 2 ) = {uv = 0}, then the (extension of the) pulled-back metric g 2 is quasi-isometric (nearby a 2 ) to the metric given by

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Re-parameterizing and reducing families of normal operators

Cited by 2 publications

References 14 publications

Multiparameter perturbation theory of matrices and linear operators

Multiparameter perturbation theory of matrices and linear operators

Monomialization of singular metrics on real surfaces

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