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July. 11th, 2024

Mini Symposium on “Determinantal Point Processes, Quantum Mechanics and Signal Analysis”

Department of Physics, Chuo University, Tokyo

Saori MORIMOTO Joint work with M.Katori and T.Shirai

Eigenvalues and Pseudospectra in Non-Hermitian and Non-Normal Matrix-Valued Processes

arXiv/math-ph/2401.08129

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Outline

1. Introduction

1.1. Classification of Matrices

1.2. Symbol Curve of Toeplitz Operators

1.3. Definition of Pseudospectrum

2. Non-normal Matrix-valued Processes

3. Theorems on Eigenvalue Processes

4. Numerical Study of Pseudospectrum Processes

5. Symbol Curve of The Two models

6. Upper Bounds of Pseudospectrum Processes

7. Future Problems

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1.1. Classification of Matrices

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Our models

 

 

generalized eigenvectors

 

 

 

Jordon Block

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1.2. Symbol Curves of Toeplitz Operators

 

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1.2. Symbol Curves of Toeplitz Operators

 

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1.2. Symbol Curves of Toeplitz Operators

 

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1.2. Symbol Curves of Toeplitz Operators

 

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Theorems for Banded Toeplitz Matrices

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Lloyd N. Trefethen, Mark Embree, `Spectra and Pseudospectra, The Behavior of Nonnormal Matrices and Operators', Princeton University Press, Princeton and Oxford, 2005

 

Proposition [ Trefethen-Embree 2005 ]

 

Theorem [ Trefethen-Embree 2005 ]

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1.3. Definition of Pseudospectrum

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Example of the resolvent norm from [1]

 

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2. Non-normal Matrix-valued Processes (Model 1)

 

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2. Non-normal Matrix-valued Processes (Model 2)

 

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Compare Model 1 and Model 2

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Model 2

 

 

Model 1

 

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Compare Model 1 and Model 2

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Model 2

 

 

Model 1

 

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3. Theorems on Eigenvalue Processes (Model1)

 

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Theorem 3.1

・・・ （3.1）

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Solution of the equation (3.1)

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Theorem 3.1

・・・ （3.1）

 

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3. Theorems on Eigenvalue Processes (Model1)

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Theorem 3.1

・・・ （3.1）

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3. Theorems on Eigenvalue Processes (Model1)

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Theorem 3.1

・・・ （3.1）

 

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3. Theorems on Eigenvalue Processes (Model1)

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Theorem 3.1

・・・ （3.1）

 

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3.1. Theorems on Eigenvalue Processes (Model1)

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Theorem 3.1

・・・ （3.1）

 

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3. Theorems on Eigenvalue processes (Model2)

 

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Theorem 3.2

・・・ （3.2）

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Compare Model 1 and Model 2

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Theorem 3.1

・・・ （3.1）

 

Theorem 3.2

・・・ （3.2）

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Solution of the equation (3.2)

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Theorem 3.2

・・・ （3.2）

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3. Theorems on Eigenvalue Processes (Model2)

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Theorem 3.2

・・・ （3.2）

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3. Theorems on Eigenvalue Processes (Model2)

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Theorem 3.2

・・・ （3.2）

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3. Theorems on Eigenvalue Processes (Model2)

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Theorem 3.2

・・・ （3.2）

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3. Theorems on The Eigenvalue Processes

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Theorem 3.2

・・・ （3.2）

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4. Numerical Study of Pseudospectrum Processes

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4.1. Eigenvalues and Pseudospectrum (Model1)

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4.1. Eigenvalues and Pseudospectrum (Model1)

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4.2. Eigenvalues and Pseudospectrum (Model2)

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4.2. Eigenvalues and Pseudospectrum (Model2)

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Conjecture 1

Numerical calculation suggests peak structures in the pseudospectrum.

More precise study is desired.

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exact eigenvalues

pseudospectrum

eigenvalue-like (peak) structures

 

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5. Symbol Curve of Model 1

 

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Symbol curves

Compare with the inner structure and symbol curve

 

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5. Symbol curve of Model 2

 

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Symbol curves

Compare with the inner structure and symbol curve

 

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Conjecture 2

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inner part of the “eigenvalues”

outermost closed simple curve

“eigenvalues”

inner part of symbol curve

exact eigenvalues

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Conjecture 2

 

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inner part of the “eigenvalues”

outermost closed simple curve

“eigenvalues”

inner part of symbol curve

exact eigenvalues

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6.1. Jordan Decomposition

We have proved the following.

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6.1. Jordan Decomposition

 

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6.2. Jordan Expansion of Resolvent

 

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6.3. Upper Bounds of Pseudospectrum Processes (Model1)

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6.3. Upper Bounds of Pseudospectrum Processes (Model1)

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7. Future Problems

 

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References

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Thank you very much for your attention.

[1] Trefethen, L. N., Embree, M.: Spectra and Pseudospectra: the Behavior of Nonnormal Matrices and Operators, vol. 1. Princeton University Press, Princeton (2005)

[2] Morimoto, S., Makoto, M., Shirai, T.: Eigenvalue and pseudospectrum processes generated by non-normal Toeplitz matrices with perturbation, arXiv/math-ph/2401.08129.

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Introduction

 

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0.1. The matrix-valued BM starts from the null matrix

 

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0.2. The matrix-valued BM starts from the shift matrix

 

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0.3. Shift matrix with Gaussian random perturbation

 

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0.4. Shift dynamics with Gaussian random perturbation

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Appearance of devil’s staircase

 

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Appearance of devil’s staircase

 

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Model 2

 

 

 

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Theorem 2.1

・・・ (２)

 

 

 

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6.3. Upper Bounds of Pseudospectrum Processes (Model1)

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