1

Makoto KATORI

(Chuo Univ, Japan)

joint work with

Saori Morimoto (Chuo)

Tomoyuki Shirai (Kyushu)

Syota Esaki (Oita)

Satoshi Yabuoku (Kitakyushu)

arXiv:math-ph/2401.08129 and math.PR/2306.00300

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A Day of Randomness

Department of Mathematics

National University of Singapore

May 24, 2024

Eigenvalues and Pseudospectra in

Nonnormal Matrix-Valued Processes

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Plan��1. Introduction

�2. Nonnormal Matrix-Valued 　　

　Dynamical Systems

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3. Numerical Observations

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4. Exact Eigenvalue Processes

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5. Conjectures for Pseudospectrum

　Processes

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6. Jordan Blocks and Generalized � Eigenvector-Overlaps

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7. Concluding Remarks

1. Introduction

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Eigenvalues vs Pseudo-Eigenvalues

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2. Nonnormal Matrix-Valued Dynamical Systems

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Model 1 and Model 2 in Morimoto-K-Shirai (2024+)

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3. Numerical Observations

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Contours of 2-norm of Resolvent

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Contours of 2-norm of Resolvent

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4. Exact Eigenvalue Processes

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Asymptotics in Infinite-Matrix Limit

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5. Conjectures for Pseudospectrum Processes

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5. Conjectures for Pseudospectrum Processes

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Scaling Property of Sizes

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6. Jordan Blocks and Generalized � Eigenvector-Overlaps

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7. Concluding Remarks

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For the non-Hermitian matrix-valued BM,

please see

arXiv:math.PR/2306.00300

(Esaki-Katori-Yabuoku)

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For the pseudospectra problem, please see

arXiv:math-ph/2401.08129

(Morimoto-Katori-Shirai)

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Thank you very much

for your attention.

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