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1

August 27th, 2024

Random Media and Random Fields, Lorentz Center @ Oort

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. Non-normal Matrix-valued Processes

2. Theorems on Eigenvalue Processes

3. Numerical Study of Pseudospectrum Processes

4. Symbol Curve of The Two models

5. Upper Bounds of Pseudospectrum Processes

6. Future Problems

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1.0. Classification of Matrices

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Our models

 

 

generalized eigenvectors

 

 

 

Jordon Block

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

 

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

 

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1.2. Compare Model 1 and Model 2

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Model 2

 

 

Model 1

 

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1.2. Defective Matrices

We will prove the following.

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2. Theorems on Eigenvalue Processes (Model1)

 

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Theorem 2.1

・・・ （2.1）

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2. Theorems on Eigenvalue processes (Model2)

 

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Theorem 2.2

・・・ （2.2）

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2. Theorems on Eigenvalue Processes

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​

​

Model 2

​

​

Model 1

 

 

We plot the solutions of Eq.(2.1) and Eq.(2.2)

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2. Theorems on Eigenvalue Processes

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​

​

Model 2

​

​

Model 1

 

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2. Theorems on Eigenvalue Processes

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​

​

Model 2

​

​

Model 1

We can find the inner structures.

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2. Theorems on Eigenvalue Processes

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​

​

Model 2

​

​

Model 1

Q1. What is this inner part?

Q2. What is this inner part structure?

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3.0. Definition of Pseudospectrum

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Example of the resolvent norm from [1]

 

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Numerical Study of Pseudospectrum Processes

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3.1. Eigenvalues and Pseudospectrum (Model1)

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3.1. Eigenvalues and Pseudospectrum (Model1)

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3.1. Eigenvalues and Pseudospectrum (Model2)

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3.1. 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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4.0. Symbol Curves of Toeplitz Operators

 

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4.1. Symbol Curve of Model 1

 

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Symbol curves

Compare with the inner structure and symbol curve

 

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4.1. 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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5.1. Jordan Decomposition

 

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5.2. Jordan Expansion of Resolvent

 

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5.3. Upper Bounds of Pseudospectrum Processes (Model1)

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Theorem 5.1

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Summary

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6. 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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