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Quantitative Diagonalizability Part I: Three Measures of Nonnormality pseudospectrum || 1 || 1 pseudospectrum || 1

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SLIDE 1

Quantitative Diagonalizability

Part I: Three Measures of Nonnormality

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SLIDE 2

πœ— βˆ’pseudospectrum

Ξ›πœ— 𝑁 ≔ 𝑨 ∈ β„‚ ∢ || 𝑨 βˆ’ 𝑁 βˆ’1|| β‰₯ πœ—βˆ’1

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SLIDE 3

πœ— βˆ’pseudospectrum

Ξ›πœ— 𝑁 ≔ 𝑨 ∈ β„‚ ∢ || 𝑨 βˆ’ 𝑁 βˆ’1|| β‰₯ πœ—βˆ’1 = {𝑨 ∈ β„‚ ∢ πœπ‘œ 𝑨 βˆ’ 𝑁 ≀ πœ— } = {𝑨 ∈ β„‚: 𝑨 ∈ π‘‘π‘žπ‘“π‘‘ 𝐡 + 𝐹 , ||𝐹|| ≀ πœ—}

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SLIDE 4

πœ— βˆ’pseudospectrum

Ξ›πœ— 𝑁 ≔ 𝑨 ∈ β„‚ ∢ || 𝑨 βˆ’ 𝑁 βˆ’1|| β‰₯ πœ—βˆ’1

For normal matrices, Ξ›πœ— 𝑁 = Ξ›0 𝑁 + 𝐸 0, πœ—

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SLIDE 5

Pseudospectrum of Toeplitz Example

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SLIDE 6

πœ— βˆ’pseudospectrum

Ξ›πœ— 𝑁 ≔ 𝑨 ∈ β„‚ ∢ || 𝑨 βˆ’ 𝑁 βˆ’1|| β‰₯ πœ—βˆ’1

e.g. discretization of pde from acoustics:

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SLIDE 7

Ξ›πœ— 𝑁 ≔ 𝑨 ∈ β„‚ ∢ || 𝑨 βˆ’ 𝑁 βˆ’1|| β‰₯ πœ—βˆ’1 = {𝑨 ∈ β„‚ ∢ πœπ‘œ 𝑨 βˆ’ 𝑁 ≀ πœ— } = 𝑨 ∈ β„‚: 𝑨 ∈ π‘‘π‘žπ‘“π‘‘ 𝐡 + 𝐹 , ||𝐹|| ≀ πœ— [Bauer-Fike]: Ξ›πœ— 𝑁 βŠ‚ Ξ›0 𝑁 + πœ†π‘“ 𝑁 𝐸 0, πœ— For distinct eigs Ξ›πœ— 𝑁 = Ξ›0 𝑁 +βˆͺ𝑗 𝐸(πœ‡π‘—, πœ† πœ‡π‘— πœ—) + 𝑝(πœ—)

πœ— βˆ’pseudospectrum

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Part II: Davies’ Conjecture

(with Jess Banks, Archit Kulkarni, Satyaki Mukherjee)

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Diagonalization

𝐡 ∈ β„‚π‘œΓ—π‘œ is diagonalizable if 𝐡 = π‘ŠπΈπ‘Šβˆ’1 for invertible π‘Š, diagonal 𝐸. Every matrix is a limit of diagonalizable matrices. Let πœ†π‘“ 𝐡 ≔ ||π‘Š|| β‹… ||π‘Šβˆ’1|| be the eigenvector condition number of 𝐡. Question: Given a matrix 𝐡 and πœ€ > 0 , what is min{πœ†π‘“ 𝐡 + 𝐹 : ||𝐹|| ≀ πœ€}?

πœ†π‘“ = ∞ πœ†π‘“ β‰ͺ ∞

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Diagonalization

𝐡 ∈ β„‚π‘œΓ—π‘œ is diagonalizable if 𝐡 = π‘ŠπΈπ‘Šβˆ’1 for invertible π‘Š, diagonal 𝐸. Every matrix is a limit of diagonalizable matrices. Let πœ†π‘“ 𝐡 ≔ ||π‘Š|| β‹… ||π‘Šβˆ’1|| be the eigenvector condition number of 𝐡. Question: Given a matrix 𝐡 and πœ€ > 0 , what is min{πœ†π‘“ 𝐡 + 𝐹 : ||𝐹|| ≀ πœ€}?

πœ†π‘“ = ∞ πœ†π‘“ β‰ͺ ∞

πœ†π‘“ 𝐡 = 1 for normal, ∞ for nondiagonalizable

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Diagonalization

𝐡 ∈ β„‚π‘œΓ—π‘œ is diagonalizable if 𝐡 = π‘ŠπΈπ‘Šβˆ’1 for invertible π‘Š, diagonal 𝐸. Every matrix is a limit of diagonalizable matrices. Let πœ†π‘“ 𝐡 ≔ ||π‘Š|| β‹… ||π‘Šβˆ’1|| be the eigenvector condition number of 𝐡. Question: Given a matrix 𝐡 and πœ€ > 0 , what is min{πœ†π‘“ 𝐡 + 𝐹 : ||𝐹|| ≀ πœ€}?

πœ†π‘“ = ∞ πœ†π‘“ β‰ͺ ∞

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  • Problem. Compute 𝑔(𝐡) for analytic function 𝑔, e.g. 𝑔 𝑨 = 𝑓𝑨, π‘¨π‘ž.

NaΓ―ve Approach. 𝑔 𝐡 = π‘Šπ‘” 𝐸 π‘Šβˆ’1. Highly unstable if πœ†π‘“(𝐡) is big.

e.g. π‘œ Γ— π‘œ Toeplitz:

Empirically: 𝐡 is close to a matrix with much better πœ†π‘“ …

Motivation: Computing Matrix Functions

πœ†π‘“ 𝐡 = 2π‘œβˆ’1 β‰ˆ 1030

πœ†π‘“(𝐡 + 𝐹) βˆ₯ 𝐹 βˆ₯

π‘œ = 100 𝐹~Gaussian

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SLIDE 13
  • Problem. Compute 𝑔(𝐡) for analytic function 𝑔, e.g. 𝑔 𝑨 = 𝑓𝑨, π‘¨π‘ž.

NaΓ―ve Approach. 𝑔 𝐡 = π‘Šπ‘” 𝐸 π‘Šβˆ’1. Highly unstable if πœ†π‘“(𝐡) is big.

e.g. π‘œ Γ— π‘œ Toeplitz:

Empirically: 𝐡 is close to a matrix with much better πœ†π‘“ …

Motivation: Computing Matrix Functions

πœ†π‘“ 𝐡 = 2π‘œβˆ’1 β‰ˆ 1030

πœ†π‘“(𝐡 + 𝐹) βˆ₯ 𝐹 βˆ₯

π‘œ = 100 𝐹~Gaussian

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SLIDE 14
  • Problem. Compute 𝑔(𝐡) for analytic function 𝑔, e.g. 𝑔 𝑨 = 𝑓𝑨, π‘¨π‘ž.

NaΓ―ve Approach. 𝑔 𝐡 = π‘Šπ‘” 𝐸 π‘Šβˆ’1. Highly unstable if πœ†π‘“(𝐡) is big.

e.g. π‘œ Γ— π‘œ Toeplitz, n=100:

Motivation: Computing Matrix Functions

πœ†π‘“ 𝐡 = 2π‘œβˆ’1 β‰ˆ 1030

πœ†π‘“(𝐡 + 𝐹) βˆ₯ 𝐹 βˆ₯

π‘œ = 100 𝐹~Gaussian

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SLIDE 15
  • Problem. Compute 𝑔(𝐡) for analytic function 𝑔, e.g. 𝑔 𝑨 = 𝑓𝑨, π‘¨π‘ž.

NaΓ―ve Approach. 𝑔 𝐡 = π‘Šπ‘” 𝐸 π‘Šβˆ’1. Highly unstable if πœ†π‘“(𝐡) is big.

e.g. π‘œ Γ— π‘œ Toeplitz, n=100:

Empirically: 𝐡 is close to a matrix with much better πœ†π‘“ …

Motivation: Computing Matrix Functions

πœ†π‘“ 𝐡 = 2π‘œβˆ’1 β‰ˆ 1030

πœ†π‘“(𝐡 + 𝐹) βˆ₯ 𝐹 βˆ₯

𝐹~Gaussian

experiment by M. Embree

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SLIDE 16
  • Problem. Compute 𝑔(𝐡) for analytic function 𝑔, e.g. 𝑔 𝑨 = 𝑓𝑨, π‘¨π‘ž.

NaΓ―ve Approach. 𝑔 𝐡 = π‘Šπ‘” 𝐸 π‘Šβˆ’1. Highly unstable if πœ†π‘“(𝐡) is big.

e.g. π‘œ Γ— π‘œ Toeplitz, n=100:

Empirically: 𝐡 is close to a matrix with much better πœ†π‘“.

Motivation: Computing Matrix Functions

πœ†π‘“ 𝐡 = 2π‘œβˆ’1 β‰ˆ 1030

πœ†π‘“(𝐡 + 𝐹) βˆ₯ 𝐹 βˆ₯

𝐹~Gaussian

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  • Idea. Approximate 𝑔(𝐡) by 𝑔 𝐡 + 𝐹 for some small 𝐹.

e.g.𝑔 𝐡 = 𝐡 E = randn(n)*delta [V,D]=eig(A+E) S = V*D.^(1/2)*inv(V)

πœ€ πœ€

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SLIDE 18
  • Idea. Approximate 𝑔(𝐡) by 𝑔 𝐡 + 𝐹 for some small 𝐹.

e.g.𝑔 𝐡 = 𝐡 E = randn(n)*delta [V,D]=eig(A+E) S = V*D.^(1/2)*inv(V)

πœ€ πœ€

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SLIDE 19
  • Idea. Approximate 𝑔(𝐡) by 𝑔 𝐡 + 𝐹 for some small 𝐹.

e.g.𝑔 𝐡 = 𝐡 E = randn(n)*delta [V,D]=eig(A+E) S = V*D.^(1/2)*inv(V)

πœ€ πœ€ experiment by M. Embree

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Approximate Diagonalization

  • Theorem. [Davies’06] For every 𝐡 ∈ β„‚π‘œΓ—π‘œ with ||𝐡|| ≀ 1 and πœ€ ∈ (0,1)

there is a perturbation 𝐹 such that πœ†π‘“ 𝐡 + 𝐹 ≀ 𝐷 π‘œ πœ€

π‘œβˆ’1

  • Conjecture. For every 𝐡 ∈ β„‚π‘œΓ—π‘œ with ||𝐡|| ≀ 1 and πœ€ ∈ (0,1) there is a

perturbation 𝐹 such that πœ†π‘“ 𝐡 + 𝐹 ≀ π·π‘œ πœ€ [Davies’06]: true for π‘œ = 3 and for special case 𝐡 = πΎπ‘œ, with π·π‘œ = 2.

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Approximate Diagonalization

  • Theorem. [Davies’06] For every 𝐡 ∈ β„‚π‘œΓ—π‘œ with ||𝐡|| ≀ 1 and πœ€ ∈ (0,1)

there is a perturbation 𝐹 such that πœ†π‘“ 𝐡 + 𝐹 ≀ 𝐷 π‘œ πœ€

π‘œβˆ’1

  • Conjecture. For every 𝐡 ∈ β„‚π‘œΓ—π‘œ with ||𝐡|| ≀ 1 and πœ€ ∈ (0,1) there is a

perturbation 𝐹 such that πœ†π‘“ 𝐡 + 𝐹 ≀ π·π‘œ πœ€ [Davies’06]: true for π‘œ = 3 and for special case 𝐡 = πΎπ‘œ, with π·π‘œ = 2.

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Approximate Diagonalization

  • Theorem. [Davies’06] For every 𝐡 ∈ β„‚π‘œΓ—π‘œ with ||𝐡|| ≀ 1 and πœ€ ∈ (0,1)

there is a perturbation 𝐹 such that πœ†π‘“ 𝐡 + 𝐹 ≀ 𝐷 π‘œ πœ€

π‘œβˆ’1

  • Conjecture. For every 𝐡 ∈ β„‚π‘œΓ—π‘œ with ||𝐡|| ≀ 1 and πœ€ ∈ (0,1) there is a

perturbation 𝐹 such that πœ†π‘“ 𝐡 + 𝐹 ≀ π·π‘œ πœ€ [Davies’06]: true for π‘œ = 3 and for special case 𝐡 = πΎπ‘œ, with π·π‘œ = 2.

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Main Result

Theorem A. For every 𝐡 ∈ β„‚π‘œΓ—π‘œ with ||𝐡|| ≀ 1 and πœ€ ∈ (0,1) there is a perturbation 𝐹 such that πœ†π‘“ 𝐡 + 𝐹 ≀ 4π‘œ3/2 πœ€ Implies every matrix has a 1/π‘žπ‘π‘šπ‘§(π‘œ) perturbation with πœ†π‘“ ≀ π‘žπ‘π‘šπ‘§(π‘œ) Implied by a stronger probabilistic result on condition number of eigenvalues.

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Main Result

Theorem A. For every 𝐡 ∈ β„‚π‘œΓ—π‘œ with ||𝐡|| ≀ 1 and πœ€ ∈ (0,1) there is a perturbation 𝐹 such that πœ†π‘“ 𝐡 + 𝐹 ≀ 4π‘œ3/2 πœ€ Implies every matrix has a 1/π‘žπ‘π‘šπ‘§(π‘œ) perturbation with πœ†π‘“ ≀ π‘žπ‘π‘šπ‘§(π‘œ) Implied by a stronger probabilistic result on condition number of eigenvalues.

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Main Result

Theorem A. For every 𝐡 ∈ β„‚π‘œΓ—π‘œ with ||𝐡|| ≀ 1 and πœ€ ∈ (0,1) there is a perturbation 𝐹 such that πœ†π‘“ 𝐡 + 𝐹 ≀ 4π‘œ3/2 πœ€ Implies every matrix has a 1/π‘žπ‘π‘šπ‘§(π‘œ) perturbation with πœ†π‘“ ≀ π‘žπ‘π‘šπ‘§(π‘œ) Implied by a stronger probabilistic result on eigenvalue condition numbers.

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Probabilistic Analysis of πœ†π‘—

Theorem B. Assume ||𝐡|| ≀ 1 and let 𝐻 have i.i.d. complex standard Gaussian entries. Let πœ‡1, … πœ‡π‘œ be the eigenvalues of 𝐡 + 𝛿𝐻.

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Probabilistic Analysis of πœ†π‘—

Theorem B. Assume ||𝐡|| ≀ 1 and let 𝐻 have i.i.d. complex standard Gaussian entries. Let πœ‡1, … πœ‡π‘œ be the eigenvalues of 𝐡 + 𝛿𝐻.

𝑨 = 𝑦 + 𝑗𝑧 where 𝑦, 𝑧~𝑂 0,

1 2

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Probabilistic Analysis of πœ†π‘—

Theorem B. Assume ||𝐡|| ≀ 1 and let 𝐻 have i.i.d. complex standard Gaussian entries. Let πœ‡1, … πœ‡π‘œ be the eigenvalues of 𝐡 + 𝛿𝐻. Then for any open ball 𝐢 βŠ‚ β„‚: 𝔽 ෍

πœ‡π‘—βˆˆπΆ

πœ†2 πœ‡π‘— ≀ π‘œ πœŒπ›Ώ2 β‹… π‘€π‘π‘š(𝐢)

πœ‡1 πœ‡2 πœ‡3 πœ‡4

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SLIDE 29

Probabilistic Analysis of πœ†π‘—

Theorem B. Assume ||𝐡|| ≀ 1 and let 𝐻 have i.i.d. complex standard Gaussian entries. Let πœ‡1, … πœ‡π‘œ be the eigenvalues of 𝐡 + 𝛿𝐻. Then for any open ball 𝐢 βŠ‚ β„‚: 𝔽 ෍

πœ‡π‘—βˆˆπΆ

πœ†2 πœ‡π‘— ≀ π‘œ πœŒπ›Ώ2 β‹… π‘€π‘π‘š(𝐢)

  • cf. Precise asymptotic results for 𝐡 = 0 [Chalker-Mehlig’98,…Bourgade-Dubach’18,Fyodorov’18]

and 𝐡 =Toeplitz [Davies-Hager’08,…Basak-Paquette-Zeitouni’14-18, Sjostrand-Vogel’18]

πœ‡1 πœ‡2 πœ‡3 πœ‡4

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SLIDE 30

Probabilistic Analysis of πœ†π‘—

Theorem B. Assume ||𝐡|| ≀ 1 and let 𝐻 have i.i.d. complex standard Gaussian entries. Let πœ‡1, … πœ‡π‘œ be the eigenvalues of 𝐡 + 𝛿𝐻. Then for any open ball 𝐢 βŠ‚ β„‚: 𝔽 ෍

πœ‡π‘—βˆˆπΆ

πœ†2 πœ‡π‘— ≀ π‘œ πœŒπ›Ώ2 β‹… π‘€π‘π‘š(𝐢)

  • cf. Precise asymptotic results for 𝐡 = 0 [Chalker-Mehlig’98,…Bourgade-Dubach’18,Fyodorov’18]

and 𝐡 =Toeplitz [Davies-Hager’08,…Basak-Paquette-Zeitouni’14-18, Sjostrand-Vogel’18] Remark: Bourgade-Dubach implies that Theorem B is sharp for 𝐡 = 0

πœ‡1 πœ‡2 πœ‡3 πœ‡4

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SLIDE 31

Implication B->A

Theorem B. Assume ||𝐡|| ≀ 1 and let 𝐻 have i.i.d. complex standard Gaussian entries. Let πœ‡1, … πœ‡π‘œ be the eigenvalues of 𝐡 + 𝛿𝐻. Then for any

  • pen ball 𝐢 βŠ‚ β„‚:

𝔽 ෍

πœ‡π‘—βˆˆπΆ

πœ†2 πœ‡π‘— ≀ π‘œ πœŒπ›Ώ2 β‹… π‘€π‘π‘š(𝐢) Proof of Theorem A. Let 𝛿 < 1/ π‘œ. whp ||𝐡 + 𝛿𝐻|| ≀ 3 so all πœ‡π‘— ∈ 𝐢 = 𝐸(0,3). πœ†π‘“ 𝐡 + 𝛿𝐻 ≀ π‘œ β‹… ෍

πœ‡π‘—βˆˆπΆ

πœ†2 πœ‡π‘— ≀ 𝑃 π‘œ 𝛿 ≀ 𝑃 π‘œ3/2 πœ€

πœ€ ≍ 𝛿 π‘œ

πœ‡1 πœ‡2 πœ‡3 πœ‡4

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SLIDE 32

Implication B->A

Theorem B. Assume ||𝐡|| ≀ 1 and let 𝐻 have i.i.d. complex standard Gaussian entries. Let πœ‡1, … πœ‡π‘œ be the eigenvalues of 𝐡 + 𝛿𝐻. Then for any

  • pen ball 𝐢 βŠ‚ β„‚:

𝔽 ෍

πœ‡π‘—βˆˆπΆ

πœ†2 πœ‡π‘— ≀ π‘œ πœŒπ›Ώ2 β‹… π‘€π‘π‘š(𝐢) Proof of Theorem A. Let 𝛿 < 1/ π‘œ. whp ||𝐡 + 𝛿𝐻|| ≀ 3 so all πœ‡π‘— ∈ 𝐢 = 𝐸(0,3). πœ†π‘“ 𝐡 + 𝛿𝐻 ≀ π‘œ β‹… ෍

πœ‡π‘—βˆˆπΆ

πœ†2 πœ‡π‘— ≀ 𝑃 π‘œ 𝛿 ≀ 𝑃 π‘œ3/2 πœ€

πœ€ ≍ 𝛿 π‘œ

πœ‡1 πœ‡2 πœ‡3 πœ‡4

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SLIDE 33

Implication B->A

Theorem B. Assume ||𝐡|| ≀ 1 and let 𝐻 have i.i.d. complex standard Gaussian entries. Let πœ‡1, … πœ‡π‘œ be the eigenvalues of 𝐡 + 𝛿𝐻. Then for any

  • pen ball 𝐢 βŠ‚ β„‚:

𝔽 ෍

πœ‡π‘—βˆˆπΆ

πœ†2 πœ‡π‘— ≀ π‘œ πœŒπ›Ώ2 β‹… π‘€π‘π‘š(𝐢) Proof of Theorem A. Let 𝛿 < 1/ π‘œ. whp ||𝐡 + 𝛿𝐻|| ≀ 3 so all πœ‡π‘— ∈ 𝐢 = 𝐸(0,3). Then with constant prob. πœ†π‘“ 𝐡 + 𝛿𝐻 ≀ π‘œ β‹… ෍

πœ‡π‘—βˆˆπΆ

πœ†2 πœ‡π‘— ≀ 𝑃 π‘œ 𝛿 ≀ 𝑃 π‘œ3/2 πœ€

πœ€ ≍ 𝛿 π‘œ

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SLIDE 34

Implication B->A

Theorem B. Assume ||𝐡|| ≀ 1 and let 𝐻 have i.i.d. complex standard Gaussian entries. Let πœ‡1, … πœ‡π‘œ be the eigenvalues of 𝐡 + 𝛿𝐻. Then for any

  • pen ball 𝐢 βŠ‚ β„‚:

𝔽 ෍

πœ‡π‘—βˆˆπΆ

πœ†2 πœ‡π‘— ≀ π‘œ πœŒπ›Ώ2 β‹… π‘€π‘π‘š(𝐢) Proof of Theorem A. Let 𝛿 < 1/ π‘œ. whp ||𝐡 + 𝛿𝐻|| ≀ 3 so all πœ‡π‘— ∈ 𝐢 = 𝐸(0,3). Then with constant prob. πœ†π‘“ 𝐡 + 𝛿𝐻 ≀ π‘œ β‹… ෍

πœ‡π‘—βˆˆπΆ

πœ†2 πœ‡π‘— ≀ 𝑃 π‘œ 𝛿 ≀ 𝑃 π‘œ3/2 πœ€

πœ€ ≍ 𝛿 π‘œ

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SLIDE 35

Proof of Theorem B

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SLIDE 36
  • 1. Area of the pseudospectrum

Lemma 1: If 𝑁 has distinct eigenvalues then for every open 𝐢: 𝜌 ෍

πœ‡π‘—βˆˆπΆ

πœ† πœ‡π‘— 2 = lim

πœ—β†’0

π‘€π‘π‘š Ξ›πœ— 𝑁 ∩ 𝐢 πœ—2

<board>

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SLIDE 37
  • 1. Area of the pseudospectrum

Lemma 1: If 𝑁 has distinct eigenvalues then for every open 𝐢: 𝜌 ෍

πœ‡π‘—βˆˆπΆ

πœ† πœ‡π‘— 2 = lim

πœ—β†’0 inf π‘€π‘π‘š Ξ›πœ— 𝑁 ∩ 𝐢

πœ—2

<board>

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SLIDE 38
  • 2. Real Anticoncentration

Theorem[Sankar-Spielman-Teng’06]: For any real π‘œ Γ— π‘œ matrix 𝑁, and G with i.i.d. real 𝑂(0,1) entries: β„™ πœπ‘œ 𝑁 + 𝛿𝐻 ≀ πœ— ≀ 𝐷 π‘œπœ—/𝛿 Proof Idea: Let 𝑁 + 𝛿𝐻 have columns 𝑛𝑗 + 𝛿𝑕𝑗, Let 𝑇 = π‘‘π‘žπ‘π‘œ 𝑛𝑗 + 𝛿𝑕𝑗 𝑗>2 β„™ 𝑒𝑗𝑑𝑒(𝑛1 + 𝛿𝑕1, 𝑇) ≀ πœ— = β„™ 𝑛1 + 𝛿𝑕1, π‘₯ ≀ πœ— = β„™ 𝑛1, π‘₯ βˆ’ 𝛿𝑕 ≀ πœ— ≀ πœ—/𝛿 𝑛1 + 𝛿𝑕1 π‘₯

Orthogonal invariance anticoncentration

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SLIDE 39
  • 2. Real Anticoncentration

Theorem[Sankar-Spielman-Teng’06]: For any real π‘œ Γ— π‘œ matrix 𝑁, and G with i.i.d. real 𝑂(0,1) entries: β„™ πœπ‘œ 𝑁 + 𝛿𝐻 ≀ πœ— ≀ 𝐷 π‘œπœ—/𝛿 Proof Idea: Let 𝑁 + 𝛿𝐻 have columns 𝑛𝑗 + 𝛿𝑕𝑗, Let 𝑇 = π‘‘π‘žπ‘π‘œ 𝑛𝑗 + 𝛿𝑕𝑗 𝑗>2 β„™ 𝑒𝑗𝑑𝑒(𝑛1 + 𝛿𝑕1, 𝑇) ≀ πœ— = β„™ 𝑛1 + 𝛿𝑕1, π‘₯ ≀ πœ— = β„™ 𝑛1, π‘₯ βˆ’ 𝛿𝑕 ≀ πœ— ≀ πœ—/𝛿 𝑛1 + 𝛿𝑕1

Orthogonal invariance anticoncentration

π‘₯

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SLIDE 40

2’. Complex Anticoncentration

Lemma 2. For any complex π‘œ Γ— π‘œ matrix 𝑁, and G with i.i.d. complex 𝑂(0,1β„‚) entries: β„™ πœπ‘œ 𝑁 + 𝛿𝐻 ≀ πœ— ≀ π‘œπœ—2/𝛿2 Proof Idea: Let 𝑁 + 𝛿𝐻 have columns 𝑛𝑗 + 𝛿𝑕𝑗, Let 𝑇 = π‘‘π‘žπ‘π‘œ 𝑛𝑗 + 𝛿𝑕𝑗 𝑗>2 β„™ 𝑒𝑗𝑑𝑒(𝑛1 + 𝛿𝑕1, 𝑇) ≀ πœ— = β„™ 𝑛1 + 𝛿𝑕1, π‘₯ ≀ πœ— = β„™ 𝑛1, π‘₯ βˆ’ 𝛿𝑕 ≀ πœ— ≀ πœ—2/𝛿2 𝑛1 + 𝛿𝑕1 π‘₯

Unitary invariance anticoncentration

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SLIDE 41

2’. Complex Anticoncentration

Lemma 2. For any complex π‘œ Γ— π‘œ matrix 𝑁, and G with i.i.d. complex 𝑂(0,1β„‚) entries: β„™ πœπ‘œ 𝑁 + 𝛿𝐻 ≀ πœ— ≀ π‘œπœ—2/𝛿2 Proof Idea: Let 𝑁 + 𝛿𝐻 have columns 𝑛𝑗 + 𝛿𝑕𝑗, Let 𝑇 = π‘‘π‘žπ‘π‘œ 𝑛𝑗 + 𝛿𝑕𝑗 𝑗>2 β„™ 𝑒𝑗𝑑𝑒(𝑛1 + 𝛿𝑕1, 𝑇) ≀ πœ— = β„™ 𝑛1 + 𝛿𝑕1, π‘₯ ≀ πœ— = β„™ 𝑛1, π‘₯ βˆ’ 𝛿𝑕 ≀ πœ— ≀ πœ—2/𝛿2 𝑛1 + 𝛿𝑕1 π‘₯

Unitary invariance anticoncentration

  • Cf. [Edelman’88] M=0
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SLIDE 42
  • 3. Expected Area of the Pseudospectrum

Lemma 2. For any complex π‘œ Γ— π‘œ matrix 𝑁, complex Gaussian 𝐻: β„™ πœπ‘œ 𝑁 + 𝛿𝐻 ≀ πœ— ≀ π‘œπœ—2/𝛿2 Applied to 𝑁 = 𝑨 βˆ’ 𝐡 βˆ’ 𝛿𝐻 says : β„™ 𝑨 ∈ Ξ›πœ— 𝐡 + 𝛿𝐻 = β„™[πœπ‘œ 𝑨 βˆ’ 𝐡 βˆ’ 𝛿𝐻 ≀ πœ—] ≀ π‘œπœ—2/𝛿2 So for every fixed ball 𝐢, for every πœ— > 0: π”½π‘€π‘π‘š Ξ›πœ— 𝐡 + 𝛿𝐻 ∩ 𝐢 = ΰΆ±

𝐢

β„™ 𝑨 ∈ Ξ›πœ— 𝐡 + 𝛿𝐻 𝑒𝑨 ≀ π‘œπœ—2 𝛿2 β‹… π‘€π‘π‘š(𝐢)

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SLIDE 43
  • 3. Expected Area of the Pseudospectrum

Lemma 2. For any complex π‘œ Γ— π‘œ matrix 𝑁, complex Gaussian 𝐻: β„™ πœπ‘œ 𝑁 + 𝛿𝐻 ≀ πœ— ≀ π‘œπœ—2/𝛿2 Lemma 3. For every fixed ball 𝐢, for every πœ— > 0: π”½π‘€π‘π‘š Ξ›πœ— 𝐡 + 𝛿𝐻 ∩ 𝐢 ≀ π‘œπœ—2 𝛿2 β‹… π‘€π‘π‘š(𝐢)

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SLIDE 44
  • 4. Expected Limiting Area of the Pseudospectrum

Define the function 𝑔

πœ— 𝐻 ≔ π‘€π‘π‘š(Ξ›πœ— 𝐡 + 𝛿𝐻 ∩ 𝐢)/πœ—2

Lemma 2 shows that lim inf

πœ—β†’0

𝔽𝑔

πœ— 𝐻 ≀ π‘œ/𝛿2

By Fatou’s lemma, 𝔽 lim inf

πœ—β†’0

𝑔

πœ— 𝐻 ≀ π‘œ/𝛿2

So by Lemma 1: 𝔽 𝜌 ෍

πœ‡π‘—βˆˆπΆ

πœ†2 πœ‡π‘— = 𝔽 lim inf

πœ—β†’ 0 𝑔 πœ—(𝐻) ≀ π‘œ

𝛿2 β‹… π‘€π‘π‘š(𝐢)

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SLIDE 45
  • 4. Expected Limiting Area of the Pseudospectrum

Define the function 𝑔

πœ— 𝐻 ≔ π‘€π‘π‘š(Ξ›πœ— 𝐡 + 𝛿𝐻 ∩ 𝐢)/πœ—2

Lemma 2 shows that lim inf

πœ—β†’0

𝔽𝑔

πœ— 𝐻 ≀ π‘œ/𝛿2

By Fatou’s lemma, 𝔽 lim inf

πœ—β†’0

𝑔

πœ— 𝐻 ≀ π‘œ/𝛿2

So by Lemma 1: 𝔽 𝜌 ෍

πœ‡π‘—βˆˆπΆ

πœ†2 πœ‡π‘— = 𝔽 lim inf

πœ—β†’ 0 𝑔 πœ—(𝐻) ≀ π‘œ

𝛿2 β‹… π‘€π‘π‘š(𝐢)

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SLIDE 46
  • 4. Expected Limiting Area of the Pseudospectrum

Define the function 𝑔

πœ— 𝐻 ≔ π‘€π‘π‘š(Ξ›πœ— 𝐡 + 𝛿𝐻 ∩ 𝐢)/πœ—2

Lemma 3 shows that lim inf

πœ—β†’0

𝔽𝑔

πœ— 𝐻 ≀ π‘œ/𝛿2

By Fatou’s lemma, 𝔽 lim inf

πœ—β†’0

𝑔

πœ— 𝐻 ≀ π‘œ/𝛿2

So by Lemma 1: 𝔽 𝜌 ෍

πœ‡π‘—βˆˆπΆ

πœ†2 πœ‡π‘— = 𝔽 lim inf

πœ—β†’ 0 𝑔 πœ—(𝐻) ≀ π‘œ

𝛿2 β‹… π‘€π‘π‘š(𝐢)

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SLIDE 47
  • 4. Expected Limiting Area of the Pseudospectrum

Define the function 𝑔

πœ— 𝐻 ≔ π‘€π‘π‘š(Ξ›πœ— 𝐡 + 𝛿𝐻 ∩ 𝐢)/πœ—2

Lemma 3 shows that lim inf

πœ—β†’0

𝔽𝑔

πœ— 𝐻 ≀ π‘œ/𝛿2

By Fatou’s lemma, 𝔽 lim inf

πœ—β†’0

𝑔

πœ— 𝐻 ≀ π‘œ/𝛿2

So by Lemma 1: 𝔽 𝜌 ෍

πœ‡π‘—βˆˆπΆ

πœ†2 πœ‡π‘— = 𝔽 lim inf

πœ—β†’ 0 𝑔 πœ—(𝐻) ≀ π‘œ

𝛿2 β‹… π‘€π‘π‘š(𝐢)

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SLIDE 48

Recap of the Proof

Let 𝑁 = 𝐡 + 𝛿𝐻 and 𝐢 = 𝐸 0,3 . 𝔽 ෍

πœ‡π‘—βˆˆπΆ

πœ† πœ‡π‘— 2 = 1 𝜌 β‹… 𝔽 lim inf

πœ—β†’0

π‘€π‘π‘š Ξ›πœ— 𝑁 ∩ 𝐢 πœ—2 ≀

1 𝜌 β‹… lim inf πœ—β†’0

𝔽

π‘€π‘π‘š Ξ›πœ— 𝑁 ∩𝐢 πœ—2

≀

9max

π‘¨βˆˆπΆ β„™ π‘¨βˆˆΞ›πœ— 𝑁

πœ—2

≀ 9π‘œ/𝛿2

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SLIDE 49

Phenomenon behind the result

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SLIDE 50

Summary and Questions

Three related notions of spectral stability (πœ†π‘“, πœ†(πœ‡π‘—), Ξ›πœ—) Can control global quantities by local singular values πœπ‘œ(𝑨 βˆ’ 𝑁) Exploited invariance and anticoncentration of complex Gaussian

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SLIDE 51

Summary and Questions

Three related notions of spectral stability (πœ†π‘“, πœ†(πœ‡π‘—), Ξ›πœ—) Can control global quantities by local singular values πœπ‘œ(𝑨 βˆ’ 𝑁) Exploited invariance and anticoncentration of complex Gaussian

  • Does a real Gaussian fail?
  • Dimension dependence in Theorem A. Dimension free bound?
  • Derandomization of the perturbation
  • Non-gaussian perturbations