Quantitative Diagonalizability Part I: Three Measures of - PowerPoint PPT Presentation

Quantitative Diagonalizability Part I: Three Measures of Nonnormality

𝜗 − pseudospectrum Λ 𝜗 𝑁 ≔ 𝑨 ∈ ℂ ∶ || 𝑨 − 𝑁 −1 || ≥ 𝜗 −1

𝜗 − pseudospectrum Λ 𝜗 𝑁 ≔ 𝑨 ∈ ℂ ∶ || 𝑨 − 𝑁 −1 || ≥ 𝜗 −1 = {𝑨 ∈ ℂ ∶ 𝜏 𝑜 𝑨 − 𝑁 ≤ 𝜗 } = {𝑨 ∈ ℂ: 𝑨 ∈ 𝑡𝑞𝑓𝑑 𝐵 + 𝐹 , ||𝐹|| ≤ 𝜗}

𝜗 − pseudospectrum Λ 𝜗 𝑁 ≔ 𝑨 ∈ ℂ ∶ || 𝑨 − 𝑁 −1 || ≥ 𝜗 −1 For normal matrices, Λ 𝜗 𝑁 = Λ 0 𝑁 + 𝐸 0, 𝜗

Pseudospectrum of Toeplitz Example

𝜗 − pseudospectrum Λ 𝜗 𝑁 ≔ 𝑨 ∈ ℂ ∶ || 𝑨 − 𝑁 −1 || ≥ 𝜗 −1 e.g. discretization of pde from acoustics:

𝜗 − pseudospectrum Λ 𝜗 𝑁 ≔ 𝑨 ∈ ℂ ∶ || 𝑨 − 𝑁 −1 || ≥ 𝜗 −1 = {𝑨 ∈ ℂ ∶ 𝜏 𝑜 𝑨 − 𝑁 ≤ 𝜗 } = 𝑨 ∈ ℂ: 𝑨 ∈ 𝑡𝑞𝑓𝑑 𝐵 + 𝐹 , ||𝐹|| ≤ 𝜗 Λ 𝜗 𝑁 ⊂ Λ 0 𝑁 + 𝜆 𝑓 𝑁 𝐸 0, 𝜗 [Bauer-Fike]: For distinct eigs Λ 𝜗 𝑁 = Λ 0 𝑁 +∪ 𝑗 𝐸(𝜇 𝑗 , 𝜆 𝜇 𝑗 𝜗) + 𝑝(𝜗)

Part II: Davies’ Conjecture (with Jess Banks, Archit Kulkarni, Satyaki Mukherjee)

Diagonalization 𝐵 ∈ ℂ 𝑜×𝑜 is diagonalizable if 𝐵 = 𝑊𝐸𝑊 −1 for invertible 𝑊 , diagonal 𝐸 . Every matrix is a limit of diagonalizable matrices. Let 𝜆 𝑓 𝐵 ≔ ||𝑊|| ⋅ ||𝑊 −1 || be the eigenvector condition number of 𝐵 . 𝜆 𝑓 ≪ ∞ 𝜆 𝑓 = ∞ Question: Given a matrix 𝐵 and 𝜀 > 0 , what is min{𝜆 𝑓 𝐵 + 𝐹 : ||𝐹|| ≤ 𝜀} ?

Diagonalization 𝐵 ∈ ℂ 𝑜×𝑜 is diagonalizable if 𝐵 = 𝑊𝐸𝑊 −1 for invertible 𝑊 , diagonal 𝐸 . Every matrix is a limit of diagonalizable matrices. Let 𝜆 𝑓 𝐵 ≔ ||𝑊|| ⋅ ||𝑊 −1 || be the eigenvector condition number of 𝐵 . 𝜆 𝑓 ≪ ∞ 𝜆 𝑓 𝐵 = 1 for normal, ∞ for nondiagonalizable 𝜆 𝑓 = ∞ Question: Given a matrix 𝐵 and 𝜀 > 0 , what is min{𝜆 𝑓 𝐵 + 𝐹 : ||𝐹|| ≤ 𝜀} ?

Diagonalization 𝐵 ∈ ℂ 𝑜×𝑜 is diagonalizable if 𝐵 = 𝑊𝐸𝑊 −1 for invertible 𝑊 , diagonal 𝐸 . Every matrix is a limit of diagonalizable matrices. Let 𝜆 𝑓 𝐵 ≔ ||𝑊|| ⋅ ||𝑊 −1 || be the eigenvector condition number of 𝐵 . 𝜆 𝑓 ≪ ∞ 𝜆 𝑓 = ∞ Question: Given a matrix 𝐵 and 𝜀 > 0 , what is min{𝜆 𝑓 𝐵 + 𝐹 : ||𝐹|| ≤ 𝜀} ?

Motivation: Computing Matrix Functions Problem. Compute 𝑔(𝐵) for analytic function 𝑔 , e.g. 𝑔 𝑨 = 𝑓 𝑨 , 𝑨 𝑞 . Naïve Approach . 𝑔 𝐵 = 𝑊𝑔 𝐸 𝑊 −1 . Highly unstable if 𝜆 𝑓 (𝐵) is big. e.g. 𝑜 × 𝑜 Toeplitz: 𝜆 𝑓 (𝐵 + 𝐹) 𝑜 = 100 𝐹~ Gaussian 𝜆 𝑓 𝐵 = 2 𝑜−1 ≈ 10 30 ∥ 𝐹 ∥ Empirically: 𝐵 is close to a matrix with much better 𝜆 𝑓 …

Motivation: Computing Matrix Functions Problem. Compute 𝑔(𝐵) for analytic function 𝑔 , e.g. 𝑔 𝑨 = 𝑓 𝑨 , 𝑨 𝑞 . Naïve Approach . 𝑔 𝐵 = 𝑊𝑔 𝐸 𝑊 −1 . Highly unstable if 𝜆 𝑓 (𝐵) is big. e.g. 𝑜 × 𝑜 Toeplitz: 𝜆 𝑓 (𝐵 + 𝐹) 𝑜 = 100 𝐹~ Gaussian 𝜆 𝑓 𝐵 = 2 𝑜−1 ≈ 10 30 ∥ 𝐹 ∥ Empirically: 𝐵 is close to a matrix with much better 𝜆 𝑓 …

Motivation: Computing Matrix Functions Problem. Compute 𝑔(𝐵) for analytic function 𝑔 , e.g. 𝑔 𝑨 = 𝑓 𝑨 , 𝑨 𝑞 . Naïve Approach . 𝑔 𝐵 = 𝑊𝑔 𝐸 𝑊 −1 . Highly unstable if 𝜆 𝑓 (𝐵) is big. e.g. 𝑜 × 𝑜 Toeplitz, n=100: 𝜆 𝑓 (𝐵 + 𝐹) 𝑜 = 100 𝐹~ Gaussian 𝜆 𝑓 𝐵 = 2 𝑜−1 ≈ 10 30 ∥ 𝐹 ∥

Motivation: Computing Matrix Functions Problem. Compute 𝑔(𝐵) for analytic function 𝑔 , e.g. 𝑔 𝑨 = 𝑓 𝑨 , 𝑨 𝑞 . Naïve Approach . 𝑔 𝐵 = 𝑊𝑔 𝐸 𝑊 −1 . Highly unstable if 𝜆 𝑓 (𝐵) is big. e.g. 𝑜 × 𝑜 Toeplitz, n=100: 𝜆 𝑓 (𝐵 + 𝐹) 𝐹~ Gaussian 𝜆 𝑓 𝐵 = 2 𝑜−1 ≈ 10 30 ∥ 𝐹 ∥ experiment by M. Embree Empirically: 𝐵 is close to a matrix with much better 𝜆 𝑓 …

Motivation: Computing Matrix Functions Problem. Compute 𝑔(𝐵) for analytic function 𝑔 , e.g. 𝑔 𝑨 = 𝑓 𝑨 , 𝑨 𝑞 . Naïve Approach . 𝑔 𝐵 = 𝑊𝑔 𝐸 𝑊 −1 . Highly unstable if 𝜆 𝑓 (𝐵) is big. e.g. 𝑜 × 𝑜 Toeplitz, n=100: 𝜆 𝑓 (𝐵 + 𝐹) 𝐹~ Gaussian 𝜆 𝑓 𝐵 = 2 𝑜−1 ≈ 10 30 ∥ 𝐹 ∥ Empirically : 𝐵 is close to a matrix with much better 𝜆 𝑓 .

Idea. Approximate 𝑔(𝐵) by 𝑔 𝐵 + 𝐹 for some small 𝐹 . e.g. 𝑔 𝐵 = 𝐵 𝜀 E = randn(n)*delta [V,D]=eig(A+E) 𝜀 S = V*D.^(1/2)*inv(V)

Idea. Approximate 𝑔(𝐵) by 𝑔 𝐵 + 𝐹 for some small 𝐹 . e.g. 𝑔 𝐵 = 𝐵 𝜀 E = randn(n)*delta [V,D]=eig(A+E) 𝜀 S = V*D.^(1/2)*inv(V)

Idea. Approximate 𝑔(𝐵) by 𝑔 𝐵 + 𝐹 for some small 𝐹 . e.g. 𝑔 𝐵 = 𝐵 𝜀 E = randn(n)*delta [V,D]=eig(A+E) 𝜀 S = V*D.^(1/2)*inv(V) experiment by M. Embree

Approximate Diagonalization Theorem. [Davies’06] For every 𝐵 ∈ ℂ 𝑜×𝑜 with ||𝐵|| ≤ 1 and 𝜀 ∈ (0,1) there is a perturbation 𝐹 such that 𝑜−1 𝑜 𝜆 𝑓 𝐵 + 𝐹 ≤ 𝐷 𝜀 Conjecture. For every 𝐵 ∈ ℂ 𝑜×𝑜 with ||𝐵|| ≤ 1 and 𝜀 ∈ (0,1) there is a perturbation 𝐹 such that 𝜆 𝑓 𝐵 + 𝐹 ≤ 𝐷 𝑜 𝜀 [Davies’06]: true for 𝑜 = 3 and for special case 𝐵 = 𝐾 𝑜 , with 𝐷 𝑜 = 2 .

Approximate Diagonalization Theorem. [Davies’06] For every 𝐵 ∈ ℂ 𝑜×𝑜 with ||𝐵|| ≤ 1 and 𝜀 ∈ (0,1) there is a perturbation 𝐹 such that 𝑜−1 𝑜 𝜆 𝑓 𝐵 + 𝐹 ≤ 𝐷 𝜀 Conjecture. For every 𝐵 ∈ ℂ 𝑜×𝑜 with ||𝐵|| ≤ 1 and 𝜀 ∈ (0,1) there is a perturbation 𝐹 such that 𝜆 𝑓 𝐵 + 𝐹 ≤ 𝐷 𝑜 𝜀 [Davies’06]: true for 𝑜 = 3 and for special case 𝐵 = 𝐾 𝑜 , with 𝐷 𝑜 = 2 .

Approximate Diagonalization Theorem. [Davies’06] For every 𝐵 ∈ ℂ 𝑜×𝑜 with ||𝐵|| ≤ 1 and 𝜀 ∈ (0,1) there is a perturbation 𝐹 such that 𝑜−1 𝑜 𝜆 𝑓 𝐵 + 𝐹 ≤ 𝐷 𝜀 Conjecture. For every 𝐵 ∈ ℂ 𝑜×𝑜 with ||𝐵|| ≤ 1 and 𝜀 ∈ (0,1) there is a perturbation 𝐹 such that 𝜆 𝑓 𝐵 + 𝐹 ≤ 𝐷 𝑜 𝜀 [Davies’06]: true for 𝑜 = 3 and for special case 𝐵 = 𝐾 𝑜 , with 𝐷 𝑜 = 2 .

Main Result Theorem A. For every 𝐵 ∈ ℂ 𝑜×𝑜 with ||𝐵|| ≤ 1 and 𝜀 ∈ (0,1) there is a perturbation 𝐹 such that 𝜆 𝑓 𝐵 + 𝐹 ≤ 4𝑜 3/2 𝜀 Implies every matrix has a 1/𝑞𝑝𝑚𝑧(𝑜) perturbation with 𝜆 𝑓 ≤ 𝑞𝑝𝑚𝑧(𝑜) Implied by a stronger probabilistic result on condition number of eigenvalues.

Main Result Theorem A. For every 𝐵 ∈ ℂ 𝑜×𝑜 with ||𝐵|| ≤ 1 and 𝜀 ∈ (0,1) there is a perturbation 𝐹 such that 𝜆 𝑓 𝐵 + 𝐹 ≤ 4𝑜 3/2 𝜀 Implies every matrix has a 1/𝑞𝑝𝑚𝑧(𝑜) perturbation with 𝜆 𝑓 ≤ 𝑞𝑝𝑚𝑧(𝑜) Implied by a stronger probabilistic result on condition number of eigenvalues.

Main Result Theorem A. For every 𝐵 ∈ ℂ 𝑜×𝑜 with ||𝐵|| ≤ 1 and 𝜀 ∈ (0,1) there is a perturbation 𝐹 such that 𝜆 𝑓 𝐵 + 𝐹 ≤ 4𝑜 3/2 𝜀 Implies every matrix has a 1/𝑞𝑝𝑚𝑧(𝑜) perturbation with 𝜆 𝑓 ≤ 𝑞𝑝𝑚𝑧(𝑜) Implied by a stronger probabilistic result on eigenvalue condition numbers.

Probabilistic Analysis of 𝜆 𝑗 Theorem B. Assume ||𝐵|| ≤ 1 and let 𝐻 have i.i.d. complex standard Gaussian entries. Let 𝜇 1 , … 𝜇 𝑜 be the eigenvalues of 𝐵 + 𝛿𝐻 .

Probabilistic Analysis of 𝜆 𝑗 Theorem B. Assume ||𝐵|| ≤ 1 and let 𝐻 have i.i.d. complex standard Gaussian entries. Let 𝜇 1 , … 𝜇 𝑜 be the eigenvalues of 𝐵 + 𝛿𝐻 . 1 𝑨 = 𝑦 + 𝑗𝑧 where 𝑦, 𝑧~𝑂 0, 2

Probabilistic Analysis of 𝜆 𝑗 Theorem B. Assume ||𝐵|| ≤ 1 and let 𝐻 have i.i.d. complex standard Gaussian entries. Let 𝜇 1 , … 𝜇 𝑜 be the eigenvalues of 𝐵 + 𝛿𝐻 . Then for any open ball 𝐶 ⊂ ℂ : 𝜇 1 𝑜 𝜆 2 𝜇 𝑗 ≤ 𝜇 3 𝔽 ෍ 𝜌𝛿 2 ⋅ 𝑤𝑝𝑚(𝐶) 𝜇 2 𝜇 4 𝜇 𝑗 ∈𝐶

Probabilistic Analysis of 𝜆 𝑗 Theorem B. Assume ||𝐵|| ≤ 1 and let 𝐻 have i.i.d. complex standard Gaussian entries. Let 𝜇 1 , … 𝜇 𝑜 be the eigenvalues of 𝐵 + 𝛿𝐻 . Then for any open ball 𝐶 ⊂ ℂ : 𝜇 1 𝑜 𝜆 2 𝜇 𝑗 ≤ 𝜇 3 𝔽 ෍ 𝜌𝛿 2 ⋅ 𝑤𝑝𝑚(𝐶) 𝜇 2 𝜇 4 𝜇 𝑗 ∈𝐶 cf. Precise asymptotic results for 𝐵 = 0 [Chalker- Mehlig’98,…Bourgade - Dubach’18,Fyodorov’18] and 𝐵 = Toeplitz [Davies- Hager’08,…Basak -Paquette- Zeitouni’14 -18, Sjostrand- Vogel’18]