1 of 47

1

Makoto KATORI

(Chuo Univ, Japan)

joint work with

Saori Morimoto (Chuo)

Tomoyuki Shirai (Kyushu)

arXiv:math-ph/2401.08129

Random Matrices and Related Topics in Jeju

Jeju Island, Korea

May 10, 2024

Eigenvalue and pseudospectrum processes

generated by nonnormal Toeplitz matrices with perturbation

2 of 47

2

Plan��1. Introduction

�2. Nonnormal Matrix-Valued 　　

　Dynamical Systems

​

3. Numerical Observations

​

4. Exact Eigenvalue Processes

​

5. Conjectures for Pseudospectrum

　Processes

​

6. Jordan Blocks and Generalized � Eigenvector-Overlaps

​

7. Concluding Remarks

3 of 47

1. Introduction

3

4 of 47

4

5 of 47

5

6 of 47

6

7 of 47

7

8 of 47

Eigenvalues vs Pseudo-Eigenvalues

8

9 of 47

9

10 of 47

10

11 of 47

11

12 of 47

12

13 of 47

13

14 of 47

14

15 of 47

15

16 of 47

16

17 of 47

17

18 of 47

18

19 of 47

2. Nonnormal Matrix-Valued Dynamical Systems

19

20 of 47

20

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

21 of 47

21

22 of 47

22

22

23 of 47

3. Numerical Observations

23

24 of 47

Contours of 2-norm of Resolvent

24

25 of 47

Contours of 2-norm of Resolvent

25

26 of 47

4. Exact Eigenvalue Processes

　　　　　　　26

27 of 47

27

28 of 47

28

29 of 47

29

30 of 47

30

31 of 47

31

32 of 47

Asymptotics in Infinite-Matrix Limit

32

33 of 47

33

34 of 47

5. Conjectures for Pseudospectrum Processes

34

35 of 47

5. Conjectures for Pseudospectrum Processes

35

36 of 47

36

37 of 47

37

38 of 47

Scaling Property of Sizes

38

39 of 47

39

40 of 47

40

41 of 47

6. Jordan Blocks and Generalized � Eigenvector-Overlaps

41

42 of 47

42

43 of 47

43

44 of 47

44

45 of 47

45

46 of 47

7. Concluding Remarks

46

47 of 47

47

​

For the non-Hermitian matrix-valued BM,

please see

arXiv:math.PR/2306.00300

(Esaki-Katori-Yabuoku)

​

For the pseudospectra problem, please see

arXiv:math-ph/2401.08129

(Morimoto-Katori-Shirai)

​

Thank you very much

for your attention.

​