APPROXIMATION OF THE ε-PSEUDOSPECTRA
OF TOEPLITZ OPERATORS
A Thesis
Presented to the
Faculty of
California State Polytechnic University, Pomona
In Partial Fulfillment
Of the Requirements for the Degree
Master of Science
In
Mathematics
By
Christine Kchech
2015 SIGNATURE PAGE
THESIS: APPROXIMATION OF THE ε-PSEUDOSPECTRA OF TOEPLITZ OPERATORS
AUTHOR: Christine Kchech
DATE SUBMITTED: Fall 2015
Department of Mathematics and Statistics
Dr. Arlo Caine Thesis Committee Chair Mathematics & Statistics
Dr. Hubertus Von Bremen Mathematics & Statistics
Dr. Randall Swift Mathematics & Statistics
ii ACKNOWLEDGMENTS
Thank you God for everything. I would like to thank my thesis adviser Dr. Caine for mentoring me throughout this process and making sure I did not give up. This paper would not have been possible if it was not for him. I would like to thank Dr. Von Bremen for always pushing me to accomplish more in my academic life as well as personal life. I would like to thank Dr. Swift for always ensuring me that I am doing well in my classes as well introducing me to Differential Equations which I grew to love. I would like to thank Dr. Rosin for not just being my professor but also my mentor. Thank you to my parents and sister for always supporting me. I would like to thank the Mathematics and
Statistics department for their support. I would also like to acknowledge the friends that I have made throughout the program for making my stay at Cal Poly Pomona worthwhile.
Lastly, I would like to thank Erik Augustine for helping me with MATLAB. The block code would not have been possible without you.
iii ABSTRACT
This paper investigates the relationship between the spectrum of a Toeplitz operator
and the spectra of its approximating Toeplitz matrices. The main tool in this investi
gation is the concept of ε-pseudo-eigenvalues introduced by Reichel and Trefethen. The
four equivalent conditions defining ε-pseudo-eigenvalues lead to four different numeri
cal computation techniques for approximating the ε-pseudospectrum. The first of these
routines generates pseudo-eigenvalues with a non-uniform statistical distribution and we
investigate these distributions in varies examples, attempting to determine whether the
ε-pseudo-eigenvalues of a large Toeplitz matrix AN concentrate near the eigenvalues of
AN.
iv Contents
List of Figures ix
1 Introduction 1
1.1 Inspiration ...... 1
1.2 Toeplitz Matrices and Operators ...... 2
1.3 The Toeplitz Theorem ...... 7
1.4 Fourier Analysis and Toeplitz Operators ...... 10
1.5 Trigonometric Moment Problem ...... 13
2 Pseudo-Eigenvalues 14
2.1 Definition of the Pseudospectrum ...... 14
2.2 Approximating the Pseudospectrum ...... 18
2.2.1 Perturbation code ...... 18
2.2.2 Unitary code ...... 23
2.2.3 Resolvent code ...... 26
2.2.4 Singular Value Decomposition code ...... 29
2.2.5 Comparing Theoretical vs. Numerical Aspects ...... 32
2.3 Reichel-Trefethen Theorems for Toeplitz operators ...... 33
v 3 Results 37
3.0.1 Triangular A ...... 37
3.0.2 Triangular B ...... 40
3.0.3 Triangular C ...... 42
3.1 Results for a General Toeplitz Matrix AN ...... 44 3.2 Conclusion ...... 47
3.3 Further Studies ...... 48
Bibiliography 55
vi List of Tables
2.1 Run times of the perturbation code for different N and ε values for
2 Toeplitz matrix AN with symbol fA(z)=z +z using perturbation code. . . 20 2.2 Run times for different N and ε values for Toeplitz matrices A with sym
2 bol fA(z)=z +z using unitary code...... 25 2.3 Run times for different N and ε values for Toeplitz matrices A with sym
2 bol fA(z)=z +z using the resolvent code...... 28 2.4 Run times for different N and ε values for Toeplitz matrices A with sym
2 bol fA(z)=z +z using singular value decomposition code...... 31
vii List of Figures
1.1 Regions and winding numbers for the Toeplitz matrix A with symbol
−1 2 3 fA(z)=2iz +z +0.7z ...... 10
2.1 Approximation of Λε (A) generated by the perturbation code for the 100×
2 100 Toeplitz matrix AN with symbol fA(z)=z +z ...... 20
2.2 Approximation of Λε (A) for 100 × 100 Toeplitz matrix A with symbol
2 fA(z)=z +z using modified perturbation code...... 22
2.3 Approximation of Λε (A) generated by the unitary code for the 100×100
2 Toeplitz matrix A with symbol fA(z)=z +z and ε = 2...... 26
2.4 Approximation of Λε (A) generated by the resolvent code for the 100 ×
2 100 Toeplitz matrix A with symbol fA(z)=z +z ...... 29
2.5 Approximation of Λε (A) generated by the singular value decomposition
2 code 100 ×100 Toeplitz matrix A with symbol fA(z)=z +z ...... 32 2.6 Approximation of Λ(A) as N increases and ε decreases...... 35
2.7 Approximation of Λε (A) with ε = 0.1 and 1000×1000 Toeplitz matrix A
2 with symbol fA(z)=z +z ...... 36
2 3.1 Approximation of Λε (A) for Toeplitz matrix A with symbol fA(z)=z+z using different N and ε values...... 39
viii Λ ( ) ( )= 1+z 3.2 Approximation of ε B for Toeplitz matrix B with symbol fB z 1−z using different N and ε values...... 41
3.3 Approximation of Λε (C) for Toeplitz matrix C with symbol fC(z)=z using different N and ε values...... 43
3.4 Approximation of Λε (A) for Toeplitz matrix A with symbol fA(z)= 2iz−1 +z2 +0.7z3 using different N and ε values...... 46
3.5 Approximation of Λε (A) and Λ(A) for Toeplitz matrix A with sym
−1 2 3 bol fA(z)=2iz + z + 0.7z using N = 500 and ε = 0.002 values where
Λε (AN ) in blue and Λ(AN ) in red...... 47
3.6 Approximation of Λε (A) for block Toeplitz matrix A using different N and ε values...... 53
ix Chapter 1
Introduction
The spectrum of a finite dimensional matrix can be computed numerically but the spectrum of an operator cannot. We will look at the spectrum of both matrices and operators as well as the connection between Fourier series and Toeplitz operators.
1.1 Inspiration
Toeplitz operators were introduced in 1910 by Toeplitz [11]. Toeplitz was studying the spectral theory of operators on infinite dimensional spaces. Toeplitz matrices and op erators have been used in many fields, such as Physics, probability theory, and function theory because their relationship to Fourier analysis. Since it is impossible to numer ically find the eigenvalues of Toeplitz operators, the idea of ε-pseudo-eigenvalues was introduced. The concept of pseudo-eigenvalues has been reinvented at least five times
[11]. The study of ε-pseudospectra of Toeplitz matrices helps to study the spectra of
Toeplitz operators, and vice versa. These connections will be explained in chapters 1 and
2.
1 1.2 Toeplitz Matrices and Operators
Definition 1.2.1. A Toeplitz matrix with coefficients (a1−N,...,a−1,a0,a1,...,aN−1) in C is a N ×N matrix of the form: ⎛ ⎞ a a ... a − a − ⎜ 0 1 N 2 N 1⎟ ⎜ ⎟ ⎜ ⎟ ⎜ a− a ⋱ ⋮ ⋮ ⎟ ⎜ 1 0 ⎟ ⎜ ⎟ = ⎜ ⎟. AN ⎜ a−2 a−1 ⋱ a1 a2 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⋮ ⋮ ⋱ a0 a1 ⎟ ⎜ ⎟ ⎝ ⎠ a1−N a2−N ... a−1 a0
N−1 ( )= ∑ k The symbol of a Toeplitz matrix AN is a complex function f defined by fAN z akz . k=1−N Recall that a linear operator on CN is a function L ∶ CN → CN such that
N 1. L(c1 +c2)=L(c1)+L(c2), for all c1,c2 ∈ C
2. L(kc)=kL(c) for all c ∈ CN and all k ∈ C.
Since matrix multiplication distributes over vector addition and commutes with scalar
multiplication, every N × N Toeplitz matrix defines a linear operator on CN. Next, we will define a Toeplitz operator, which is an infinite Toeplitz matrix acting on a space of
∞ 2 sequences (an) = in C. The space l (N) is the vector space of all infinite sequences n 0 √ ( , ,...) ∞ ∣ ∣2 ∥( , ,...)∥ = ∞ ∣ ∣2 a0 a1 such that ∑n=0 an converges, with norm a0 a1 2 ∑n=0 an .
( )∞ C Definition 1.2.2. A Toeplitz operator A with coefficients ak k=−∞ in is a linear oper ator on l2(N) given by multiplication by the infinite matrix
⎛ a a a ...⎞ ⎜ 0 1 2 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜a− a a ⋱ ⎟ = ⎜ 1 0 1 ⎟. A ⎜ ⎟ ⎜ ⎟ ⎜a−2 a−1 a0 ⋱ ⎟ ⎜ ⎟ ⎝ ⋮ ⋱ ⋱ ⋱ ⎠
2 ∞ k The symbol of the Toeplitz operator is the complex function fA(z)= ∑ akz . Note that k=−∞ the sequence of coefficients of such a matrix must decay in order for multiplication to
2 k preserve l (N) and for the series ∑akz to define a function. The symbol f (z) belongs in the Wiener class if the sequence of coefficients is absolutely summable.
We write S1 for the unit circle in C and L2(S1) for the Hilbert space of functions on
1 1 ( ) ( ) 2( 1) S with the inner product 2πi ∫S1 f z g z dz. Then L S + denotes the subspace of the boundary values of functions analytic in the open unit disk D.
( )= ∞ n Lemma 1.2.3. If the symbol fA z ∑n=0 anz belongs to the Wiener class then A is bounded on l2(N).
Proof. We will provide some idea on the proof but refer to Widom [9] for the full argu
( )= ∞ n , , 2 ,... ment. Suppose A is a Toeplitz operator with symbol fA z ∑n=0 anz . Since 1 z z 2 1 2 is an orthogonal basis for L (S )+ then multiplication by A on l (N) is equivalent to 2( 1) ∈ 2(N) ( )= ∞ n multiplication by the symbol fA on L S +. That is if x l and g z ∑n=0 xnz ∥ ∥ =∥ ∥ 1 then Ax l2(N) fAg L2(S1)+ .If fA is a bounded function on S , then 1 ∥ ∥ =( ∣ ( ) ( )∣2 ) 2 fAg L2(S1)+ ∫ fA z g z dz S1 1 2 2 2 =(∫ ∣ fA(z)∣ ∣g(z)∣ dz) S1 1 2 2 2 ≤(∫ ∥ fA∥∞∣g(z)∣ dz) S1 1 2 2 =∥ fA∥∞ (∫ ∣g(z)∣ dz) S1
=∥ ∥∞∥ ∥ fA g L2(S1)+ =∥ ∥ ∥ ∥ . fA ∞ x l2(N)
∥ ∥ ≤∥ ∥ ∥ ∥ Therefore, Ax l2(N) fA ∞ x l2(N) when fA is a bounded function, but ∞ ∞ ∞ ∞ n n n ∥ fA∥∞ =∥∑ anz ∥∞ ≤ ∑ ∥anz ∥∞ =∑ ∣an∣∥z ∥∞ = ∑ ∣an∣. n=0 n=0 n=0 n=0
3 The idea is that if fA is in the Wiener class, then fA is a bounded function and therefore A is a bounded operator on l2(N). A main result of Fourier analysis is that a function is continuous on S1 (and therefore bounded) if its sequence of Fourier coefficients are absolutely summable.
Definition 1.2.4. Let T ∶ l2(N)→l2(N) be a linear operator. With T , we associate the operator = −λ Tλ T I
λ 2(N) where is a complex number and I is the identity operator on l .IfT λ has an inverse, ( ) we denote it by Rλ T . That is,
( )= −1 =( −λ )−1 Rλ T Tλ T I
and it is called the resolvent operator of T at λ or simply the resolvent of T .
Definition 1.2.5. Let T ∶ l2(N)→l2(N) be a linear operator. A regular value λ of T is a complex number such that:
( ) 1. Rλ T exists,
( ) 2. Rλ T is bounded,
( ) 2(N) 3. Rλ T is defined on a set which is dense in l .
The resolvent set ρ(T ) of T is the set of all regular values λ of T . Its complement
Λ(T )=C−ρ(T ) in the complex plane C is called the spectrum of T , and each λ ∈ Λ(T ) is called a spectral value of T .
The spectrum for a finite dimensional matrix vs. an infinite dimensional operator can have completely different characteristics. For a Toeplitz matrix AN , the spectrum
4 consists of only eigenvalues and is a finite set. The eigenvalues of AN are values of λ ( ) such that Rλ A does not exist. For a Toeplitz operator A, the spectrum not only ( ) consists of eigenvalues but of other values too. This is because Rλ A can exist but not be bounded. The following two examples illustrate Definitions 1.2.4 and 1.2.5.
2 2 Example 1.2.6. Define A2 ∶ C → C by multiplication by the matrix
⎛ ⎞ ⎜31⎟ A2 = ⎜ ⎟. ⎝13⎠
=( −λ ) To find the resolvent, we will use Definition 1.2.4. We need to evaluate Tλ A2 I . Thus, ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ 31 λ 0 3 −λ 1 = ⎜ ⎟−⎜ ⎟ = ⎜ ⎟. Tλ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎝13⎠ ⎝0 λ ⎠ ⎝ 13−λ ⎠ To find the inverse, we will use the standard 2×2 matrix formula, obtaining
⎛−λ − ⎞ ⎛−λ − ⎞ 1 ⎜3 1 ⎟ 1 ⎜3 1 ⎟ Rλ (A )= ⎜ ⎟ = ⎜ ⎟ 2 (3 −λ )2 −12 (λ −4)(λ −2) ⎝ −13− λ ⎠ ⎝ −13− λ ⎠ provided λ ≠ 2,4. Thus, the resolvent set ρ(A2) of A2 is then the set
ρ(A2)=C −{2,4}
and the spectrum is Λ(A2)={2,4}.
Example 1.2.7. Define A ∶ l2(N)→l2(N) by multiplication by the matrix
⎛010 0 ⋯⎞ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜001 0 ⋱ ⎟ = ⎜ ⎟. A ⎜ ⎟ ⎜ ⎟ ⎜000 1 ⋱ ⎟ ⎜ ⎟ ⎝ ⋮ ⋱ ⋱ ⎠
5 Then A is a Toeplitz operator with symbol f (z)=z. We will show R0(A) does not exist. When λ = 0, the operator A −λ I = A is not invertible because
⎛x ⎞ ⎛0 ⎞ ⎛ ⎞ ⎜ 0⎟ ⎜ ⎟ 010 ⋯ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎜x1⎟ ⎜0⎟ = ⎜ ⎟⎜ ⎟ = ⎜ ⎟ Ax ⎜001 ⋱⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎜x2⎟ ⎜0⎟ ⎝ ⋮ ⋱ ⋱⎠⎜ ⎟ ⎜ ⎟ ⎝ ⋮ ⎠ ⎝ ⋮ ⎠
t is solved by 0 = x1 = x2 = ... but x0 is free. Thus, x =(1,0,0,...) is in the null space of A. Therefore, 0 ∈ Λ(A). Next, we will show Λ(A)=Δ where Δ is the closed unit disk in C λ = ∈ Λ( ) λ ≠ ( ) . We know that 0 A . Let 0. We need to show Rλ A exists and is bounded if and only if λ ∉ Δ, i.e, ∣λ ∣>1. Let
⎛ −1 −1 −1 ...⎞ ⎜ λ (λ )2 (λ )3 ⎟ ⎜ ⎟ ⎜ −1 −1 ⎟ ⎜ 0 ...⎟ ( )=⎜ λ (λ )2 ⎟ Rλ A ⎜ ⎟ ⎜ −1 ...⎟ ⎜ 0 0 λ ⎟ ⎜ ⎟ ⎝ ⋮ ⋮ ⋮ ⋱ ⎠ ( )( −λ )= then we will show that Rλ A A I I. By inspection,
− − − ⎛ 1 1 1 ...⎞⎛−λ 100... ⎞ ⎛1000... ⎞ ⎜ λ (λ )2 (λ )3 ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜ −1 −1 ...⎟⎜ −λ ...⎟ ⎜ ...⎟ ⎜ 0 ⎟ ⎜ 0 10 ⎟ ⎜0100 ⎟ ⎜ λ (λ )2 ⎟⎜ ⎟ = ⎜ ⎟. ⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜ −1 ...⎟⎜ −λ ...⎟ ⎜ ...⎟ ⎜ 0 0 λ ⎟ ⎜ 00 1 ⎟ ⎜0010 ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎝ ⋮ ⋮ ⋮ ⋱ ⎠⎝ ⋮ ⋮ ⋮ ⋮ ⋱ ⎠ ⎝ ⋮⋮⋮⋮⋱⎠ ( ) λ ≠ ( ) Thus, Rλ A exists for 0. We need to determine when Rλ A is bounded. Ob ∞ 1 k ( ) ( )=∑ (− + ) serve that Rλ A is also a Toeplitz operator with symbol f z k=0 λ k 1 z . Thus by ∞ 1 ( ) ( ) ∑ ∣− + ∣= Lemma 1.2.3, Rλ A is bounded if f z belongs to the Wiener class. But k=0 λ k 1 k+1 ∑∞ ( 1 ) 1 < ⇔∣λ∣> ( ) k=0 ∣λ ∣ is a geometric series and ∣λ ∣ 1 1. Therefore, Rλ A is bounded when λ > 1. It is also clearly unbounded when ∣λ ∣≤1 as one can see by applying A to
( , 1, 1 , 1 ,...) ∈ 2(N) Λ( )=Δ 1 2 3 4 l in that case. Thus A .
6 1.3 The Toeplitz Theorem
N−1 ( )= ∑ k Theorem 1.3.1. Let AN be a triangular Toeplitz matrix with symbol fAN z akz . k=0 N N Then AN defines a linear operator AN ∶ C → C by matrix multiplication and Λ(AN )=
{a0}.
N−1 × ( )= ∑ k Proof. Let AN be a N N Toeplitz matrix with symbol fAN z akz . Then A is k=0 triangular and
⎛ ⎞ ⎛ ⎞ ⎛ ⎞ a a a ... a − λ 00... 0 a −λ a a ... a − ⎜ 0 1 2 N 1⎟ ⎜ ⎟ ⎜ 0 1 2 N 1 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ a a ⋱ ⋮ ⎟ ⎜ λ 0 ⋱⋮⎟ ⎜ a −λ a ⋱ ⋮ ⎟ ⎜ 0 1 ⎟ ⎜ ⎟ ⎜ 0 1 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ −λ = ⎜ ⎟−⎜ ⎟ = ⎜ ⎟ A I ⎜ a0 ⋱ a2 ⎟ ⎜ λ ⋱ 0 ⎟ ⎜ a0 −λ ⋱ a2 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⋱ ⎟ ⎜ ⋱ ⎟ ⎜ ⋱ ⎟ ⎜ a1 ⎟ ⎜ 0 ⎟ ⎜ a1 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ a0 λ a0 −λ is upper triangular. Thus the determinant of A−λ I is just the product of the main diagonal
N entries. That is, det(A −λ I)=(a0 −λ ) . Thus, Λ(A)={a0}.
Next, we consider the spectrum for a triangular Toeplitz operator. This result was
discovered by Toeplitz in the early 1900’s [3].
∞ k Theorem 1.3.2. Let A be a triangular Toeplitz operator with symbol f (z)= ∑ akz in k=0 the Wiener class. Then Λ(A)= f (Δ) where Δ is the closed unit disk in C.
∞ k Proof. Let A be a Toeplitz operator with symbol f (z)= ∑ akz in the Wiener class. For k=0 any z ∈ D, where D is the open unit disk in the complex plane, the vector (1,z,z2 ,...)t
2(N) ∞ ∣ n∣2 = ∞ ∣ 2∣n belongs to l because ∑n=0 z ∑n=0 z is a convergent geometric series when ∣z∣<1. We will show how x =(1,z,z2 ,...)t is an eigenvector of A with eigenvalue f (z),
7 i.e., Ax = f (z)x. ⎡ ⎤ ⎢ ⎥ ⎡ ⎤⎢ 1 ⎥ ⎢ ⎥⎢ ⎥ ⎢a a a ... a − ...⎥ ⎢ ⎥ ⎡ ⎤ ⎢ 0 1 2 N 1 ⎥⎢ ⎥ ⎢ − ⎥ ⎢ ⎥⎢ z⎥ ⎢ a +a z +a z2 +...+a − zN 1 +...⎥ ⎢ ⎥⎢ ⎥ ⎢ 0 1 2 N 1 ⎥ ⎢ a0 a1 a2 ⋱⋱⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ 2 ⎥ ⎢ 2 N−1 ⎥ ⎢ ⎥⎢ z ⎥ ⎢ a0z +a1z +...+aN−2z +... ⎥ = ⎢ ⎥⎢ ⎥ = ⎢ ⎥ Ax ⎢ a0 a1 a2 ⋱ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ 2 3 N−1 ⎥ ⎢ ⎥⎢ ⋮ ⎥ ⎢ a0z +a1z +...+aN−3z +... ⎥ ⎢ ⋱ ⎥⎢ ⎥ ⎢ ⎥ ⎢ a0 a1 ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ N−1 ⎥ ⎢ ⎥ ⎢ ⎥⎢z ⎥ ⎣ ⋮ ⎦ ⎢ ⎥⎢ ⎥ ⎣ ⋱⋱⎦⎢ ⎥ ⎢ ⎥ ⎣ ⋮ ⎦ ⎡ ⎤ ⎢ ⎥ ⎢ 1 ⎥ ⎡ ∞ ⎤ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ∑ a zk ⎥ ⎢ ⎥ ⎢ k ⎥ ⎢ z ⎥ ⎢ k=0 ⎥ ⎢ ⎥ ⎢ ∞ ⎥ ⎢ ⎥ ⎢ k+1 ⎥ ⎢ ⎥ ⎢ ∑ akz ⎥ ∞ ⎢z2 ⎥ = ⎢k=0 ⎥ =(∑ k)⎢ ⎥ = ( ) . ⎢ ∞ ⎥ akz ⎢ ⎥ f z x ⎢ + ⎥ = ⎢ ⎥ ⎢ ∑ k 2 ⎥ k 0 ⎢ ⋮ ⎥ ⎢ akz ⎥ ⎢ ⎥ ⎢k=0 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ N ⎥ ⎢ ⎥ ⎢z ⎥ ⎣ ⋮ ⎦ ⎢ ⎥ ⎢ ⎥ ⎢ ⋮ ⎥ ⎣ ⎦ This implies that Λ(A) includes f (D).Now,Λ (A) is closed because the spectrum of an
operator is always closed [4]. By a property of closure, Λ(A)=Λ(A)⊇ f (D). Since f
is analytic on D and bounded on Δ because it belongs to the Wiener class, f is a closed
mapping. Thus f (D) = f (D)= f (Δ). Thus, Λ(A)⊇ f (Δ). To prove f (Δ)⊇Λ(A), we will
prove C− f (Δ)⊆C−Λ(A). Note that C−Λ(A) is the resolvent set of A. Let λ ∈ C− f (Δ).
We will show (λ I −A)−1 exists, is bounded, and is densely defined. For existence, define ( )= 1 ( ) g z λ − f (z) . Then g z is analytic near zero and
a2 ( )= 1 = 1 + a1 +( a2 + 1 ) 2 +.... g z 2 2 z 2 3 z λ −a0 −a1z −a2z −... λ −a0 (λ −a0) (λ −a0) (λ −a0)
, ,... ( ) = 1 , = a1 , =a2 + If b0 b1 denote the Taylor coefficients of g z (i.e. b0 λ− b1 2 b2 2 a0 (λ −a0) (λ −a0)
8 2 a1 , 3 and so on) then, (λ −a0) ⎡ ⎤⎡ ⎤ ⎡ ⎤ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢b b b ⋯⎥ ⎢λ −a −a −a ⋯⎥ ⎢100 ⋯⎥ ⎢ 0 1 2 ⎥⎢ 0 1 2 ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ = ⎢ ⎥ ⎢ b0 b1 ⋯⎥ ⎢ λ −a0 −a1 ⋯⎥ ⎢ 10⋯ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⋱⋱⎥⎢ ⋱⋱⎥ ⎢ ⋱⋱⎥ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎡ ⎤ ⎢ ⎥ ⎢b b b ⋯⎥ ⎢ 0 1 2 ⎥ ⎢ ⎥ = ⎢ ⎥ (λ − )−1 λ ∈ C − (Δ) because gf 1. Therefore ⎢ b0 b1 ⋯⎥ must be I A . Since f , the ⎢ ⎥ ⎢ ⎥ ⎢ ⋱⋱⎥ ⎣ ⎦ ( )= 1 Δ (λ − )−1 function g z λ − f (z) is bounded on and therefore I A is a bounded operator on l2(N). Therefore, λ ∈ C −Λ(A). Hence C − f (Δ)⊆C −Λ(A).
The Toeplitz theorem characterizes the spectrum of a triangular Toeplitz operator as the image of Δ under its symbol, which is an analytic function. For more general Toeplitz operators, the symbol is not analytic on Δ and yet there is another characterization of the spectrum.
For a piecewise smooth closed curve Γ, the winding number of Γ about a point P is −( − ) +( − ) (Γ, )= 1 y y0 dx x x0 dy =( , ) defined by the line integral I P π ∫Γ 2 2 where P x0 y0 . When 2 (x−x0) +(y−y0) Γ = ( 1) λ = + = + 1 1 f S , x0 iy0, and w x iy then this integral is equivalent to 2πi ∫ f (S) w−λ dw. ′ Δ (Γ,λ)= 1 f ()z If f is analytic on D and continuous on then we get I 2πi ∫Γ f (z)−λ dz with w = f (z).So f (Δ)= f (S1)∪{λ ∈ C ∶ I( f (S1),λ )≠0}.
∞ k Theorem 1.3.3. Let A be a Toeplitz operator with symbol f (z)= ∑ akz in the Wiener k=−∞ class. Then, Λ(A)= f (S1)∪{λ ∈ C ∶ I( f (S1),λ )≠0} with
′ 1 ( ) 1 1 f z 1 I( f() S ,λ )= ∫ dz,λ ∈/ f (S ). 2πi S f (z)−λ
Proof. Refer to Reichel and Trefethen [1].
9 −1 2 3 Example 1.3.4. Consider the Toeplitz matrix A with the symbol fA(z)=2iz +z +0.7z . The curve f (S1) is shown in Figure 1.1. The winding number for each horn is +1 and
the winding number for the face is −1.
Figure 1.1: Regions and winding numbers for the Toeplitz matrix A with symbol fA(z)= 2iz−1 +z2 +0.7z3.
1.4 Fourier Analysis and Toeplitz Operators
In this section, we see the relationship between Toeplitz operators and Fourier anal ysis. A Fourier series is an expansion of a periodic function f ∶[−π,π]→R and has the form ∞ a0 f (θ)∼ + ∑ (an cos(nθ)+bn sin(nθ )) (1.1) 2 n=1
10 where
π = 1 (θ) (θ) a0 ∫ f d π −π π = 1 (θ) ( θ) (θ) ≥ an ∫ f cos n d n 1 π −π π = 1 (θ) ( θ) (θ) ≥ . bn ∫ f sin n d n 1 π −π
These expansions are very useful for solving linear partial differential equations. Using
θ − θ iθ = (θ)+ (θ) ( θ)= ein +e in ( θ)= Euler’s formula, e cos isin , we can write cos n 2 and sin n θ − θ ein −e in ( θ) ( θ) 2i . Substituting for cos n and sin n into (1.1), we get
∞ inθ −inθ inθ −inθ a0 e +e e −e f (θ)∼ + ∑ (an +bn ). (1.2) 2 n=1 2 2i
Combining terms and splitting the sum into two sums of einθ and e−inθ gives
∞ ∞ 1 −inθ 1 1 inθ f (θ)∼∑ (an +ibn)e+ a0 + ∑ (an −ibn)e . (1.3) n=1 2 2n=1 2
Changing the index for e−inθ terms, (1.3) becomes
−1 ∞ 1 inθ 1 1 inθ f (θ)∼ ∑ (a−n +ib−n)e + a0 + ∑ (an −ibn)e . (1.4) n=−∞ 2 2n=1 2 = 1 ( + ) < = 1 ( − ) > = 1 Let cn 2 a−n ib−n for n 0 as well as cn 2 an ibn for n 0 and c0 2 a0. Then (1.4) becomes
−1 ∞ (iθ )n (iθ )0 (iθ)n (iθ)n f (θ)∼ ∑ (cne )+c0e + ∑(cne )=∑ cne , (1.5) n=−∞ n=1 n∈Z
where c0 = c¯0 and c−n = c¯n for all n ≥ 1. Thus, (1.5) can be written as one sum as
∞ n g(z)∼ ∑ cnz , (1.6) n=−∞
11 iθ iθ where z = e is on the unit circle in C. Thus, f (θ )=g(z) with z = e . To compute cn, we need to plug the terms an and bn into cn. This gives for n > 0,
1 c = (a −ib ) n 2 n n 1 1 π 1 π = ( ∫ f (θ)cos(nθ)dθ −i ∫ f (θ )sin(nθ)dθ) 2 π −π π −π 1 π = ∫ f (θ)(cos(nθ)−isin(nθ))dθ 2π −π π 1 −inθ = ∫ f (θ)e dθ 2π −π
iθ iθ dz dz (θ)= ( ) = = θ θ = = θ Since f g z and z e , dz ie d implies iei iz d . Thus, ( ) = 1 g z . cn ∮ + dz 2iπ S zn 1
Since g(z)= f (θ ), the computation of the Fourier series of f (θ) is equivalent to computing the Laurent series centered at 0 of g(z) where z = eiθ . The infinite matrix
⎛⋱ ⋰ ⎞ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ a− a a ⎟ ⎜ 1 0 1 ⎟ ⎜ ⎟ = ⎜ ⎟ C ⎜ a−1 a0 a1 ⎟ (1.7) ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ a−1 a0 a1 ⎟ ⎜ ⎟ ⎝⋰ ⋱ ⋱ ⋱ ⎠
built from the coefficients of g is called a Laurent operator on l2(Z). Here, we show that multiplication by g on L2(S1) is equivalent to multiplication by C on l2(Z). Indeed, if
( )= n h z ∑n∈Z xnz then
m n k k g(z)h(z)=(∑ cmz )(∑ xnz )= ∑ ( ∑ cmxn)z = ∑ (∑ ck−mxn)z m∈Z n∈Z m∈Z m+n=k k∈Z n∈Z
12 is the symbol of the composition of Laurent operators
⎛⋱ ⋰ ⎞⎛⋱ ⋰ ⎞ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ x− x x ⎟⎜ a− a a ⎟ ⎜ 1 0 1 ⎟⎜ 1 0 1 ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟. ⎜ x−1 x0 x1 ⎟⎜ a−1 a0 a1 ⎟ (1.8) ⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ x−1 x0 x1 ⎟⎜ a−1 a0 a1 ⎟ ⎜ ⎟⎜ ⎟ ⎝⋰ ⋱ ⋱ ⋱ ⎠⎝⋰ ⋱ ⋱ ⋱ ⎠
The Toeplitz operator associated to C is the compression of this to the Hardy space
L(S1)+. Observe that the lower right block of (1.7) is a Toeplitz operator on l2(N).
1.5 Trigonometric Moment Problem
One application of Toeplitz matrices and operators is to the trigonometric moment problem. Let α(x) be a distribution function in [−π,π] and
π = 1 −inπ α( ), ∈ Z cn ∫ e d x n (1.9) 2π −π be the trigonometric moments of α. Given a sequence (cn)n∈Z, for there to exist a dis
tribution α such that (1.9) holds true, conditions on the complex sequence (cn)n∈Z need to be met. Since c¯n = c−n because α is real valued, we only need to consider the se ( )∞ ( ) quence cn n=0. In order for a given sequence cn to be the trigonometric moments of some distribution function α(x), certain additional conditions on the sequence must be met. According to Grenander and Szego in [3], each of the Toeplitz matrices AN for ( )= n the Toeplitz operator with symbol f z ∑n∈Z cnz must have positive determinant. The
determinant of AN is positive if Λ(AN )⊆R≥0. Since Λ(AN )⊆Λ(A) and Λ(A) can be determined with the Toeplitz theorems, Λ(AN )⊆R≥0 can be determined by deciding if
Λ(A)⊆R≥0. This is one reason that one is interested in the spectra of Toeplitz operators.
13 Chapter 2
Pseudo-Eigenvalues
2.1 Definition of the Pseudospectrum
N×N Given A ∈ C we write ∥A∥ for the operator norm of AN, i.e,
∥Ax∥ ∥A∥=sup{ 2 ∣ 0 ≠ x ∈ CN}. ∥x∥2
Theorem 2.1.1. Let A ∈ CN×N, i.e., A is a N × N matrix. Given ε > 0, the following
conditions on λ ∈ C are equivalent:
i. λ is an eigenvalue of A +E for some E ∈ CN×N with ∥E∥≤ε;
N ii. ∃u ∈ C ,∥u∥2 = 1, such that ∥(λ I −A)u∥2 ≤ ε;
iii. ∥(λ I −A)−1∥≥ε−1;
iv. σmin(λ I −A)≤ε, where σmin denotes the smallest singular value.
Proof. Let A ∈ CN×N and ε > 0 be given. To prove the equivalence we will show i ⇒ iii, iii ⇔ iv, iv ⇒ ii and ii ⇒ i.
14 i ⇒ iii: Suppose λ is an eigenvalue of A + E for some E ∈ CN×N with ∥E∥≤ε.By the definition there exists an x ≠ 0 ∈ CN such that (A + E)x = λ x. Then Ax + Ex = λ x which implies Ex = λ x − Ax and thus Ex =(λ I − A)x.Ifλ is not an eigenvalue of A
then λ I − A is invertible so we can multiply by (λ I − A)−1. This gives (λ I − A)−1Ex =
(λ I −A)−1(λ I −A)x which reduces to (λ I −A)−1Ex = x. By rescaling x if necessary we
may assume ∥x∥=1. Then
1( =∥x∥=∥ λ I −A)−1Ex∥≤∥(λ I −A)−1∥∥Ex∥≤∥(λ I −A)−1∥∥E∥∥x∥. (2.1)
Since ∥x∥=1 and ∥E∥≤ε, inequality (2.1) gives 1 ≤∥(λ I −A)−1∥ε which is equivalent to iii). If λ is an eigenvalue of A then λ I −A is singular, so condition iii will hold vacuously.
iii ⇔ iv: Suppose λ ∈ C and ε > 0. We want to show that σmin(λ I −A)≤ε if and only ∥(λ − )−1∥≥ε−1 σ ( )= 1 if I A . From the theory of singular values [8], min B −1 and σmax(B )
σmax(B)=∥B∥ for all matrices B. Thus if M = λ I −A then
σ (λ − )=σ ( )= 1= 1 = 1 [ ]. min I A min M −1 −1 −1 8 (2.2) σmax(M ) ∥M ∥ ∥(λ I −A) ∥
−1 −1 Thus σmin(λ I −A)≤ε if and only if ∥(λ I −A) ∥≥ε .
iv ⇒ ii: Suppose σmin(λ I −A)≤ε. Suppose λ is an eigenvalue of A. By the definition there exists an x ≠ 0 ∈ CN such that Ax = λ x. Then (λ I −A)x = λ x−Ax = λ x−λ x = 0. Thus
∥(λ I −A)x∥2 =∥0∥2 = 0 ≤ ε. Let M = λ I −A and suppose λ is not an eigenvalue of A. Then
N ∗ 2 there exist u ∈ C with ∥u∥2 = 1 such that M Mu = σ u where σ = σmin(λ I −A)≤ε. Since λ is not an eigenvalue of A, M is invertible. Thus M∗ is invertible and
Mu = σ 2(M∗)−1u. (2.3)
Applying the 2-norm to (2.3),
2 ∗ −1 2 ∗ −1 2 ∗ −1 ∥Mu∥2 =∥σ (M ) u∥2 = σ ∥(M ) u∥2 ≤ σ ∥(M ) ∥∥u∥2. (2.4)
15 ∗ −1 ∗ −1 1 1 1 Now ∥(M ) ∥=σmax((M ) )= ∗ = = σ by properties of singular value σmin(M ) σmin(M) [8]. Thus, 2.4 is equivalent to
1 ∥Mu∥ ≤ σ 2 ∥u∥ = σ ≤ ε. (2.5) 2 σ 2
N ii ⇒ i: Let ε > 0 and suppose there exists u ∈ C , ∥u∥=1 such that ∥(λ I −A)u∥2 ≤ ε.
N×N Since u is a unit vector then there exists a unitary matrix Q ∈ C such that Qu = e1 = [100 ...0]T . We want to construct a matrix E ∈ CN×N such that λ is an eigenvalue of
A +E and ∥E∥≤ε. Set
= ∗ [ ∗ ] . E Q (λ I −QAQ )e1 ∣ 0 ∣⋯∣ 0 Q
Then
(A +E)u =Au +Eu
= ∗ +( ∗ [ ∗ ] ) ∗ AQ e1 Q (λ I −QAQ )e1 ∣ 0 ∣⋯∣ 0 Q Q e1
∗ ∗ ∗ =AQ e1 +(Q (λ I −QAQ )e1)
∗ ∗ ∗ =AQ e1 +(λ Q e1 −AQ e1)
∗ =λ Q e1
=λ u.
Thus λ is an eigenvalue of A+E with eigenvector u. Next we will check if ∥E∥≤ε. Since
16 √ ∗ ∥E∥= σmax(E)= max-eigenvalue(E E), and ⎡ ⎤ ⎢ ∗ ∗⎥ ⎢((λ I −QAQ )e ) ⎥ ⎢ 1 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ 0 ⎥ ∗ = ∗ ⎢ ⎥ ∗ [ ∗ ] E E Q ⎢ ⎥QQ ((λ I − QAQ )e1) 0 ⋯ 0 Q ⎢ ⎥ ⎢ ⋮ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ 0 ⎦ ⎡ ⎤ ⎢ ∗ ⎥ ⎢∥((λ I − QAQ )e )∥2 0 ⋯ 0 ⎥ ⎢ 1 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ 0 0 ⋯ 0 ⎥ = ∗ ⎢ ⎥ Q ⎢ ⎥Q ⎢ ⎥ ⎢ ⋮ ⋱ ⋮ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⋯ ⎥ ⎣ 0 0⎦ ∗ ∗ 2 we see that E E has eigenvalues 0 and ∥(λ I − QAQ )e1∥ and thus √ ∗ ∥E∥= σmax(E)= max-eigenvalue(E E) √ = ∥(λ − ∗) ∥2 I QAQ e1 2
∗ =∥(λ I −A)Q e1∥2
=∥(λ I −A)u∥2 ≤ ε.
Definition 2.1.2. Given ε > 0, a number λ ∈ C is an ε-pseudo-eigenvalue of A if any of the following equivalent conditions are satisfied:
i. λ is an eigenvalue of A +E for some E ∈ CN×Nwith ∥E∥≤ ε;
N ii. There exist u ∈ C ,∥u∥2 = 1, such that ∥(A − λ I)u∥2 ≤ ε;
iii. ∥(λ I − A)−1∥≥ ε−1;
iv. σmin(λ I − A)≤ ε.
The set of all ε-pseudo-eigenvalues of A, the ε-pseudospectrum of A, is denoted by
Λε (A).
17 2.2 Approximating the Pseudospectrum
⎛ ⎞ ⎜31 ⎟ Example 2.2.1. We recall the example A2 = ⎜ ⎟. The spectrum is two points Λ(A2)= ⎝13⎠
{2,4}. To find the ε-pseudospectrum of A2, we use condition four. Since the matrix is
2×2, we are able to determine the singular values of λ I −A2 explicitly using the quadratic
formula, finding Λε (A2)={z ∶∣z −2∣<ε or∣z −4∣<ε}. When ε = 1, the pseudospectrum is a union of two disjoint disks. When ε > 1, the two disks intersect to form a more
complicated set. The point is that Λε (A2) is a thickening of Λ(A2) depending on ε.
We were able to find the ε-pseudospectrum explicitly in this example because N = 2, but when N is larger, we need to use MATLAB to approximate the ε-pseudospectrum.
In this section, we will look at the four equivalent conditions defining an ε-pseudo eigenvalue numerically.
2.2.1 Perturbation code
We use condition one to estimate Λε (AN) by sampling random N ×N matrices con sisting of complex numbers such that the operator norm is less than a fixed epsilon. We
run over fifty random matrices of size N. We then plot the image f (S1) under the symbol
f of the Toeplitz matrix AN along with the eigenvalues of AN +E.
n=100; %size of matrix
r =[0 ,1 ,1 ,0 ,0]; %row of matrix
c=[0 ,0 ,0 ,0 ,0]; %column of matrix
for k=1:n−5
r=[r ,0];
c=[c ,0];
18 end
A=toeplitz (c,r); %Toeplitz matrix
yy =[]; %empty array
for i =1:50
b=0.01; %epsilon
G= randn (n,n)+ j * randn(n,n); %random matrix
E=b*G./max(svd(G)); %rescaled to have operator norm b lambda=eig (A+E); %eigenvalues of A+E
yy=[yy;lambda]; %stores all the lambdas
end
t=linspace (0,2* pi );
z=cos(t)+j* sin ( t ); u=z+z .ˆ(2); %matrix symbol
plot(yy, ’o’) %plots eigenvalues
grid on
hold on
plot(u) %plots symbol
The machine that was used to run the codes has a 2 GHz processor with 4 GB RAM.
The system is not fast due to the amount of memory already used. To get a better time the matrices should be run on a desktop. The following table shows the time in seconds of running the perturbation code for different ε and N values for Toeplitz matrices with
2 symbol fA(z)=z +z .
19 N = 50 N = 100 N = 500
ε = 0.02 1.6 seconds 6.8 seconds 496.3 seconds
ε = 0.01 2 seconds 7 seconds 545 seconds
ε = 0.002 1.6 seconds 6.7 seconds 621.6 seconds
Table 2.1: Run times of the perturbation code for different N and ε values for Toeplitz
2 matrix AN with symbol fA(z)=z +z using perturbation code.
The time did increase as N increased as we expected. However, for ε = 0.002 the time for N = 50 and N = 100 was smaller then the previous two cases and was larger than the two for N = 500.
Example 2.2.2. We obtain the approximation of Λε (AN ) shown in figure 2.2 for a 100 ×
2 100 matrix A with ε = 0.01 for the symbol fA(z)=z +z .
Figure 2.1: Approximation of Λε (A) generated by the perturbation code for the 100×100
2 Toeplitz matrix AN with symbol fA(z)=z +z .
From figure 2.1 we can see that most of the pseudo-eigenvalues are lying around a
1 limacon close to fA(S ) with some scattered in the interior. The densest clustering of the pseudo-eigenvalues from this sampling occurs near the complex numbers -1 and 0.
20 In the previous code, we were sampling matrices E with a fixed epsilon, i.e. ∥E∥=ε.
To ensure that we sample E′ s with ∥E∥≤ε we add a for loop to sample over E′ s with
∥E∥ equal to different epsilon values.
clear ;
clc
close all
n=500;
yy =[];
r =[0 ,1 ,1 ,0 ,0];
c=[0 ,0 ,0 ,0 ,0];
for k=1:n−5
r=[r ,0];
c=[c ,0];
end m=0.002;
for b=0.01*m:10ˆ( − 1)*m:m; %epsilon decreasing for i =1:50
A=toeplitz (c,r); %Toeplitz matrix
G=randn (n,n)+ j* randn(n,n); %random matrix
E=b*G./max(svd(G)); %rescaled to have operator norm b lambda=eig (A+E); %eigenvalues
yy=[yy ; lambda ];
end
end
t=linspace (0,2* pi );
21 z=cos(t)+j* sin ( t ); u=z+z .ˆ(2); %symbol
plot (yy , ’o’)
grid on
hold on
plot (u)
s=real (yy );
q=imag (yy );
d=[s ,q];
figure (2);
hist3 (d)
N=hist3 (d);
xlabel ( ’ real ’)
ylabel ( ’ Imaginary ’)
Example 2.2.3. With the modified perturbation code for a 100 × 100 Toeplitz matrix A
2 symbol fA(z)=z +z . We obtain the approximation to Λε (A) shown in figure 2.3.
Figure 2.2: Approximation of Λε (A) for 100×100 Toeplitz matrix A with symbol fA(z)= z +z2 using modified perturbation code.
22 From figure 2.2 we can see the pseudo-eigenvalues are clustered together around the limacon and inner loop. However, more pseudo-eigenvalues are scattered in the interior.
Using different epsilon values gives a better approximation of Λε . We suspect that the most clustering occurs near -1 and 0.
2.2.2 Unitary code
Condition two requires that we find a vector u, such that ∥u∥2 = 1 satisfying ∥(λ I −
A)u∥≤ ε to determine whether λ ∈ Λε (A). Since we need to find a vector satisfying the condition, we run a hundred random vectors that need to be normalized. The lambdas
that are picked to determine whether the condition holds are selected from a grid we set
up. Each value in the grid must be tested. If the value satisfies the condition, then we
plot it; if not, we move on. In order for the code to run, we found it necessary to set ε ≥ 2
to ensure that some λ ′ s can be computed.
n=100; %size of matrix
t=linspace (0,2* pi );
z=cos ( t )+1 j * sin ( t ); u=z+z .ˆ2; %symbol of matrix
r =[0 ,1 ,1 ,0 ,0]; %row of matrix
c=[0 ,0 ,0 ,0 ,0]; %column of matrix
for k=1:n−5
r=[r ,0];
c=[c ,0];
end
A=toeplitz (c,r); %Toeplitz matrix
e=2; %epsilon
23 K=40; %number of horizontal subdivisions
L=40; %number of vertical subdivisions
xmin=min( real (u ));
xmax=max( real (u));
ymin=min( imag (u ));
ymax=max( imag (u ));
y =[]; %empty array
for k=1:K
for l =1:L
lambda=(xmin +((xmax−xmin )/K)* k)+1 j *(ymin +((ymax−ymin )/L)* l); for s=1:10
v=randn (n ,1);
h=v ./ norm (v );
roww=norm (((A−lambda* eye (n ,n))* h )); i f roww<=e
y= [y; real ( lambda ) , imag ( lambda ) ] ; %stores lambdas
end
end
end
end
r=y(: ,1);
im=y(: ,2);
scatter (r ,im)
hold on
plot (u)
24 The following table shows the time in seconds of running the unitary code for differ
2 ent ε and N values for Toeplitz matrices with symbol fA(z)=z +z .
N = 50 N = 100 N = 500
ε = 2.5 361.4 seconds 401.5 seconds 1647.8 seconds
ε = 2.25 261.1 seconds 301.8 seconds 1651.1 seconds
ε = 2 134.6 seconds 185.5 seconds 1749.3 seconds
Table 2.2: Run times for different N and ε values for Toeplitz matrices A with symbol
2 fA(z)=z +z using unitary code.
Run times for the unitary code depend on the size of the matrix, epsilon value, and the number of vectors being tested. In our case, a 100 random vectors are tested searching for a vector satisfying the condition. When ε decreases, the time increases for N = 500 but decreases for N = 50 and N = 100 although the accuracy of the simulation diminishes.
Searching for a unit vector u such that ∥(λ I −A)u∥2 ≤ ε is clearly not very efficient and, furthermore, fails to label tested λ ′ s as pseudo-eigenvalues even when ε = 1.
Example 2.2.4. ε-pseudo-eigenvalues for a 100 ×100 Toeplitz matrix A and with ε = 2,
2 detected by the unitary code with the symbol fA(z)=z +z .
25 Figure 2.3: Approximation of Λε (A) generated by the unitary code for the 100 × 100
2 Toeplitz matrix A with symbol fA(z)=z +z and ε = 2.
In figure 2.3 we can see that the unitary code registers pseudo-eigenvalues that are both inside and outside the limacon. In part, this is because the value of ε is large, but is also due to the difficult nature of the search for a qualifying unit vector. We observe that the pseudo-eigenvalues are mostly scattered to the left. The spread of the pseudo- eigenvalues depends on the grid size, the number of random unit vectors tested, as well as the epsilon and matrix size.
2.2.3 Resolvent code
Condition three is similar to condition two in that we set up a grid and test the points in the grid for membership in Λε (A). However, the difference is that we do not need a unit vector but instead we evaluate the 2-norm of (λ I −A)−1. This can cause MATLAB to crash if (λ I −A)−1 is close to singular.
n=100;
t=linspace (0,2* pi );
z=cos ( t )+1 j * sin ( t );
26 u=z+z .ˆ2; %symbol of matrix
for k=1:n−5
r =[0 ,1 ,1 ,0 ,0]; %row of matrix
c=[0 ,0 ,0 ,0 ,0]; %column of matrix
end
A=toeplitz (c,r); %Toeplitz matrix
e=.01; %epsilon
K=40; %number of horizontal subdivisions
L=40; %number of vertical subdivisions
xmin=min( real (u ) ) ;
xmax=max( real (u));
ymin=min( imag (u ) ) ;
ymax=max( imag (u ) ) ;
y =[]; %empty array
for k=1:K
for l=1:L
lambda=(xmin +((xmax−xmin )/K)* k)+1j *(ymin +((ymax−ymin )/L)* l);
boc=norm (( inv ( lambda* eye (n ,n)−A))); if boc>=eˆ−1
y= [y; real ( lambda ) , imag ( lambda ) ] ; %stores lambdas
end
end
end
r=y(: ,1);
im=y(: ,2);
27 scatter (r ,im)
hold on
plot (u ) , axis ([ xmin xmax ymin ymax ])
The following table shows the run times in seconds for the resolvent code for different
2 ε and N values for Toeplitz matrices with symbol fA(z)=z +z .
N = 50 N = 100 N = 500
ε = 0.02 10.2 seconds 42.4 seconds 4240.7 seconds
ε = 0.01 9.1 seconds 37.7 seconds 4311.1 seconds
ε = 0.002 9.4 seconds 37.5 seconds 4416.3 seconds
Table 2.3: Run times for different N and ε values for Toeplitz matrices A with symbol
2 fA(z)=z +z using the resolvent code.
The time for each N varies as epsilon decreases. As epsilon decreases and N increases we expect the time to increase. This is not the case when ε = 0.01 and N = 100. It takes approximately an hour to run the cases for N = 500. In general, MATLAB takes longer to run when computing the inverse for larger matrices.
Example 2.2.5. Results of the resolvent code for a 100 ×100 Toeplitz matrix A with the
2 symbol fA(z)=z +z for ε = 0.01.
28 Figure 2.4: Approximation of Λε (A) generated by the resolvent code for the 100 × 100
2 Toeplitz matrix A with symbol fA(z)=z +z .
Figure 2.4 shows that the pseudo-eigenvalues identified by the resolvent code are nearly filling up the limacon. The identified pseudo-eigenvalues seem to overlap the interior loop of the limacon, but they do not overlap the outside loop. In contrast to the unitary code, this important accuracy, but it comes at a cost in run times for large matrices.
2.2.4 Singular Value Decomposition code
Condition four is similar to condition two and three in that we set up a grid and test values λ to see if σmin(λ I − A)≤ε. We use the MATLAB command “min(svd)” to calculate the smallest singular value of (λ I −A).
n=100;
t=linspace (0,2* pi );
z=cos ( t )+1 j * sin ( t ); u=z+z .ˆ2; %matrix symbol
r =[0 ,1 ,1 ,0 ,0]; %row of matrix
29 c=[0 ,0 ,0 ,0 ,0]; %column of matrix
for k=1:n−5
r=[r ,0];
c=[c ,0];
end
A=toeplitz (c,r); %Toeplitz matrix
e=.01; %epsilon
K=40;
L=40;
xmin=min( real (u ));
xmax=max( real (u));
ymin=min( imag (u ));
ymax=max( imag (u ));
y=[];
for k=1:K
for l =1:L
lambda=(xmin +((xmax−xmin )/K)* k)+1 j *(ymin +((ymax−ymin )/L)* l);
sigmaN=min( svd ( lambda* eye (n , n)−A)); i f sigmaN<=e
y= [y; real ( lambda ) , imag ( lambda ) ] ;
end
end
end
r=y(: ,1);
im=y(: ,2);
30 scatter (r ,im)
hold on
plot (u ) , axis ([ xmin xmax ymin ymax ])
The following table shows the run times in seconds for running the singular value
decomposition code for different ε and N values for Toeplitz matrices with symbol
2 fA(z)=z +z .
N = 50 N = 100 N = 500
ε = 0.02 3.9 seconds 14.4 seconds 2113.8 seconds
ε = 0.01 4.1 seconds 14.1 seconds 2237.2 seconds
ε = 0.002 4.2 seconds 15.5 seconds 2159.8 seconds
Table 2.4: Run times for different N and ε values for Toeplitz matrices A with symbol
2 fA(z)=z +z using singular value decomposition code.
As indicated by the table in figure 2.8 this code takes longer to run compared to the perturbation code. For N = 50, the time increases as epsilon decreases. The time does not significantly change with decreasing epsilon at N = 100.
Example 2.2.6. Results of the singular value decomposition code for a 100×100 Toeplitz
2 matrix A with symbol fA(z)=z +z and ε = 0.01.
31 Figure 2.5: Approximation of Λε (A) generated by the singular value decomposition code
2 100 ×100 Toeplitz matrix A with symbol fA(z)=z +z .
Figure 2.5 shows the pseudo-eigenvalues are within the limacon without any lying on the solid line. The distribution is also uniform. There are spots inside the limacon where pseudo-eigenvalues have not been identified.
2.2.5 Comparing Theoretical vs. Numerical Aspects
We have shown the four conditions defining ε-pseudo-eigenvalues of a matrix A to be
theoretically equivalent but not numerically equivalent in the sense that our codes imple
menting the four conditions produced different approximations of Λε (A) on our exam ple. With the perturbation code, we observed that the distribution of computed pseudo- eigenvalues is not uniform. Furthermore, the pseudo-eigenvalues are mostly around the limacon and its interior loop. The unitary code computed pseudo-eigenvalues outside of the limacon as well as inside and the distribution depended on the number of unitary vectors being tested as well as the epsilon value. The resolvent code produced a uni form distribution, but the code crashed if (λ I − A)−1 was close to singular. Lastly, the
singular value decomposition code produced a uniform distribution with all computed
pseudo-eigenvalues inside the limacon as expected. This gives evidence that for large N,
32 Λε (AN)≈ f (Δ). This suggests that Λε (AN )≈Λ(A) for large N and small ε.
2.3 Reichel-Trefethen Theorems for Toeplitz operators
The observation that Λε (AN)≈Λ(A) for large N and small ε inspire [1] to study
Λε (A) has a proxy for Λ(A).
( )= ∞ k Definition 2.3.1. Let A be a non-diagonal Toeplitz operator with symbol fA z ∑k=0 akz
in the Wiener class. Write fN (z) for the symbol of the associated N ×N Toeplitz matrix and define cN and r by N−1 cN = ∑ ∣ak∣>0 (2.6) k=1 1 ε N r =( ) . (2.7) cN
( )= ∞ k Theorem 2.3.2. Let A be a non-diagonal Toeplitz operator with symbol fA z ∑k=0 akz . Then with notation as in Definition 2.7
fN (Δr)⊆Λε (AN )⊆ fN(Δ)+Δε (2.8) where Δr denotes the disclosed unit disk of radius r [1].
Proof. See Reichel and Trefethen [1], page 164.
The following theorem from Reichel and Trefethen [1], page 163 explains how the
ε-pseudospectrum of a triangular Toeplitz matrix AN approximates ε-pseudospectrum of a triangular Toeplitz operator.
Theorem 2.3.3. Let A be a triangular Toeplitz operator with symbol f (z) in the Wiener class. Then
lim Λε (AN)= f (Δ)+Δε = Λε (A) (2.9) N→∞
33 for each ε > 0, and therefore
lim lim Λε (AN)= f (Δ)=Λ(A). (2.10) ε→0 N→∞
Proof. Let ε > 0 be given. We will make use of the fact that limN→∞ Λε (AN)=Λε (A) which was proved by Widom in [10]. First, since f is in the Wiener class, the coefficients 1 (a ) are absolutely summable and the sequence (c ) is bounded. Hence r =( ε ) N con k N cN verges to 1 as N →∞. Since
fN (Δr)⊆Λε (AN )⊆ fN(Δ)+Δε
for all N by Theorem 2.3.2, we obtain
f (Δ)⊆ lim Λε (AN)⊆ f (Δ)+Δε N→∞
′ in the limit as N →∞. Next, we will show that f (Δ)+Δε ⊆ Λε (A). Suppose λ ∈
′ f (Δ)+Δε . Then λ = λ + λ0 where λ ∈ f (Δ) and ∣λ0∣≤ε. By the Toeplitz theorem, ′ f (Δ)=Λ(A).IfE = λ0I then ∥E∥=∥λ0I∥=∣λ0∣≤ε and thus λ = λ + λ0 ∈ Λ(A + E) by ′ construction. Therefore λ ∈ Λε (A) by definition. Hence f (Δ)+Δε ⊆ Λε (A). By Widom’s result, Λε (A)=limN→∞ Λε (AN ) and thus
f (Δ)+Δε ⊆ Λε (A)= lim Λε (AN )⊆ f (Δ)+Δε N→∞ implies that
f (Δ)+Δε = Λε (A) as was to be shown. Taking the limit as ε → 0 and using Toeplitz theorem gives
f (Δ)=lim lim Λε (AN ) ε→0 N→∞ and therefore
Λ(A)=lim lim Λε (AN ). ε→0 N→∞
34 To study the spectrum of a Toeplitz operator A, Reichel and Trefethen approximated the ε-pseudospectrum of a Toeplitz matrix AN for large N and small ε, using the pertur bation code. We wish to determine if the distribution of Λε (AN) concentrates near the spectrum of approximating Toeplitz matrices AN .
ε → 0 Λε (AN) Λ(AN)
(2.3.3) N →∞ N →∞
Λε (A) Λ(A)= f (Δ) ε → 0
Figure 2.6: Approximation of Λ(A) as N increases and ε decreases.
The situation is summarized in Figure 2.6. The limit established in Theorem 2.3.3 is illustrated on the diagonal, but is not the same as first taking ε → 0 and then N →
∞. This is clear from triangular examples such as the Toeplitz operator with symbol
2 f (z)=z + z . The question is then to what extent Λ(AN) is representation of Λ(A) for
N large. Furthermore, does this distribution of Λε (AN) produced through perturbation
concentrate near Λ(AN) for N large?
Example 2.3.4. We first consider the symbol f (z)=z +z2 with N = 1000 and ε = 0.01.
35 Figure 2.7: Approximation of Λε (A) with ε = 0.1 and 1000×1000 Toeplitz matrix A with
2 symbol fA(z)=z +z .
Figure 2.7 shows ε-pseudo-eigenvalues are outside the red line which is the r value from equation (2.7). The pseudospectrum Λε (A) is approximately the image of disk
Δr = Δ1. The larger the N the more ε-pseudo-eigenvalues are going to be outside of ( 1) →∞ ε → Λ ( )=Λ( ) ε → f Sr . Taking the limit as both N and 0, the ε AN A . Note that as 0,
Λε (AN)→Λ(AN ) which consists of the single point at 0, so Λ(AN ) does not converge to Λ(A)= f (Δ) as N →∞. However, Theorem 2.3.3 says Λε (AN ) does converge to Λ(A) if we first take N →∞and then ε → 0 afterwards. Thus two different results occur depending on which limit gets taken first and if the limit of N and ε gets taken together.
36 Chapter 3
Results
To investigate the concentration of the spectrum of a Toeplitz operator, we look at the
ε-pseudospectrum generated by the perturbation code for approximating Toeplitz matri
ces for three characteristically different symbols. Given that Λε (AN), for ε small and N large, is a good approximation to Λ(A), we imagine this distribution as an approximation of the density of states function for the operator A. We would like to know whether this distribution concentrates near Λ(AN ) for N large and ε small. In other words, to what extent to the spectrum of AN, for N large, reflect the spectrum of A?
3.0.1 Triangular A
2 Consider the Toeplitz operator A with symbol fA(z)=z +z and the Toeplitx matrix
⎛0110⎞ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 011⎟ =⎜ ⎟ ( = ). AN ⎜ ⎟ N 4 ⎜ ⎟ ⎜ 01⎟ ⎜ ⎟ ⎝ 0⎠
37 Since matrix A is upper triangular, Λ(AN)={0} for all N ∈ N. To look at the ε-pseudo eigenvalues, the perturbation code will be applied with N = 50,100,500 and ε = 0.02,0.01,0.002.
38 ε = 0.02
N = 50 N = 100 N = 500
ε = 0.01
N = 50 N = 100 N = 500
ε = 0.002
N = 50 N = 100 N = 500
2 Figure 3.1: Approximation of Λε (A) for Toeplitz matrix A with symbol fA(z)=z + z ε using different N and values. 39 Figure 3.1 shows the different cases for N and ε. When ε = 0.02 and N was increas
ing, the pseudo-eigenvalues started to spread to the limacon curve. The mode is occurring
near the interior loop and the intersection between the outer and interior loop with corre
sponding values 0 and -1. The values occurring at or near -1 have a multiplicity two while
the values occurring at or near 0 have a multiplicity of one. The pseudo-eigenvalues with
multiplicity two do not make a difference because they can be considered to only have a
multiplicity of one. The smaller ε got the farther the pseudo-eigenvalues were from the
2 interior loop as well as the outer loop. Thus Λε (A)=0 for the symbol fA(z)=z +z .
3.0.2 Triangular B
( )= 1+z Consider the Toeplitz operator B with symbol fB z 1−z and the Toeplitx matrix
⎛1222⎞ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 122⎟ =⎜ ⎟ ( = ) BN ⎜ ⎟ N 4 ⎜ ⎟ ⎜ 12⎟ ⎜ ⎟ ⎝ 1⎠
The spectrum of B is Λ(BN )={1} for all N ∈ N. Note that fB is not a bounded function on
Δ, having a pole at z = 1. Thus fB is not in the Wiener class. Nonetheless, the relationship
between Λε (AN ) and Λ(A) is still valid. We thought it would be interesting to analyze the concentration of the spectrum in this case in place of triangular examples with large op
erator norm. In this case Λ(B)={z ∈ C∣Re(z)>0}. To look at the ε-pseudo-eigenvalues,
condition 1 will be applied with N = 50,100,500 and ε = 10(−20),10(−30),10(−40).
40 ε = 10(−20)
N = 50 N = 100 N = 500
ε = 10(−30)
N = 50 N = 100 N = 500
ε = 10(−40)
N = 50 N = 100 N = 500
Λ ( ) ( )= 1+z Figure 3.2: Approximation of ε B for Toeplitz matrix B with symbol fB z 1−z using 41 different N and ε values. Figure 3.2 shows the different cases for N and ε. For the case ε = 10(−20), the pseudo- eigenvalues grew wider as N increases and the pseudo-eigenvalues started to converge to the left side of the ellipse near one. The histograms indicate that when N = 50, the pseudo-eigenvalues were almost spread out evenly around the ellipse but as N increases to 500, the mode is concentrated between −i and i.Asε decreases to 10(−30), the ellipse got smaller but similar results regarding the histogram and pseudo-eigenvalue as the previous epsilon case. The results are similar for N = 50 and ε = 10(−40). However, the
results differ for N = 100,500. An inner loop of pseudo-eigenvalues is inside the ellipse.
In this case, Λε (BN) concentrates well away from Λ(BN)={1} in contrast to the previous case.
3.0.3 Triangular C
Consider the Toeplitz operator B with symbol fC(z)=z and the Toeplitz matrix
⎛0100⎞ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 010⎟ = ⎜ ⎟ ( = ). CN ⎜ ⎟ N 4 ⎜ ⎟ ⎜ 01⎟ ⎜ ⎟ ⎝ 0⎠
The spectrum of C is Λ(CN)={0} for all N ∈ N. To look at the ε-pseudo-eigenvalues, perturbation code will be applied with N = 50,100,500 and ε = 0.02,0.01,0.002.
42 ε = 0.02
N = 50 N = 100 N = 500
ε = 0.01
N = 50 N = 100 N = 500
ε = 0.002
N = 50 N = 100 N = 500
Figure 3.3: Approximation of Λε (C) for Toeplitz matrix C with symbol fC(z)=z using ε different N and values. 43 Figure 3.3 shows different cases for N and ε. In the case when ε = 0.02, the pseudo-
eigenvalues are growing towards the solid unit disk as N increases. In this case, the pseudo-eigenvalues are mainly around the disk of pseudo-eigenvalues. The center has only a few pseudo-eigenvalues. The histogram of the three N values show the mode is occurring in four places which are the diagonal corners of the unit disk. Also as N increases the pseudo-eigenvalues distribute almost uniformly around the edges of the unit disk. By decreasing epsilon to 0.01 and 0.002, similar results occur except the pseudo-eigenvalues fill in more of the disk for larger N. The histograms also indicate the distribution is concentrating in four corners as well. The matrix C has an infinite condition number which is bad because the eigenvalues are then sensitive to perturbation as we can see. In this example, Λε (CN ) does not at all concentrate near Λ(CN )={0}.
3.1 Results for a General Toeplitz Matrix AN
Reichel and Trefethen observed that Theorem 2.3.3 also hold for general Toepliz matrices and operators. They investigated Λε (AN) for large N and small ε.
Theorem 3.1.1. Let A be an arbitrary Toeplitz operator with symbol fA(z) om the Wiener
class, and let AN denote its N ×N Toeplitz matrix section. Then
lim Λε (AN)=Λε (A) N→∞
for each ε > 0, and therefore
lim lim Λε (AN )=Λ(A). ε→0 N→∞
Proof. The proof is a result from a corollary of theorem II by H.Widom’s paper, “On the singular values of Toeplitz matrices.”
44 The theorem states that even for a non-triangular Toeplitz operator A, the set Λε (AN ) converges to Λ(A) as N →∞and then ε → 0. But for non-triangular A, the spectrum of
AN is much richer and the operator is less sensitive to perturbation. So we investigate
whether Λ(AN ) resembles Λ(A) in such as same.
45 ε = 0.02
N = 50 N = 100 N = 500
ε = 0.01
N = 50 N = 100 N = 500
ε = 0.002
N = 50 N = 100 N = 500
−1 Figure 3.4: Approximation of Λε (A) for Toeplitz matrix A with symbol fA(z)=2iz + 2 + . 3 ε z 0 7z using different N and values. 46 By applying Theorem 1.3.3, we can characterize Λ(A) as the set of points in C about which f (s) has a non-zero winding number. This example was shown in Figure 1.1. The set has the shape of a bull face with two horns where the horns have a winding number +1 and the face has a winding number −1. The pseudo-eigenvalues that are connecting the two horns and face are insensitive to perturbations according to Reichel and Trefethen.
This is because the matrix AN has a small condition for each N.AsN increases and ε decreases, the pseudo-eigenvalues of A are filing out Λ(A) as expected. Interestingly, the set Λ(AN) for N large is more reflective of Λ(A) but does not appear to fill out the horns or the interior of the face. The histograms appear to show the spectrum concentrating
near the largest magnitude eigenvalues of AN as well.
Figure 3.5: Approximation of Λε (A) and Λ(A) for Toeplitz matrix A with symbol fA(z)=
−1 2 3 2iz + z + 0.7z using N = 500 and ε = 0.002 values where Λε (AN) in blue and Λ(AN ) in red.
3.2 Conclusion
We have looked at three characteristically different symbols to make a decision if the distribution of the spectrum of a Toeplitz matrices concentrate near the spectrum of approximating Toeplitz operator’s? Using the perturbation code as well as the theorems
47 from chapter one and two, we can conclude the answer is yes and no. For the symbol,
2 fA(z)=z + z , we were able to see the ε-pseudo-eigenvalues do concentrate near the = 1+z ε eigenvalues of the Toeplitz matrix A. However, for the symbol fB 1−z the -pseudo eigenvalues was very near the eigenvalues of the Toeplitz matrix B. Lastly, for the symbol
f (z)=z, the ε-pseudo-eigenvalues did concentrate near the eigenvalues of the Toeplitz
matrix C. We can also conclude the order in which the limits are taken in Reichel and
Trefethen Theorem 2.3.3 does matter. We were able to get different results depending
on whether we took N →∞or ε → 0 first or concurrently. We also applied Reichel and
Trefethen to non-triangular Toeplitz matrices in which Λε (AN)=Λ(A) for N large and ε small. The results for non-triangular Toeplitz matrices were much better than triangular
Toeplitz matrices such that the spectrum of Toeplitz matrices does concentrate near the
spectrum of Toeplitz operators for non-triangular Toeplitz operators.
3.3 Further Studies
We have looked at the triangular Toeplitz matrix and the general Toeplitz matrix. The next case would be the 2 ×2 block Toeplitz matrix. The 2 ×2 block Toeplitz matrix is a matrices whose entries are constant along block diagonals. The form for an N ×N matrix
Ais ⎡ ⎤ ⎢ ⎥ ⎢ A A A ... A − ...⎥ ⎢ 0 1 2 N 1 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ A− A A ... A − ⋱ ⎥ ⎢ 1 0 1 N 2 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⋮ ⋱ ⋱ ⋱ ⋮ ⋱ ⎥ = ⎢ ⎥ A ⎢ ⎥ ⎢ ⎥ ⎢A2−N ⋯ A−1 A0 A1 ⋱ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⋯⋯ ⋱ ⎥ ⎢A1−N A −1 A0 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⋮ ⋮ ⋮ ⋮ ⋮ ⋱ ⎥ ⎣ ⎦
48 where each Ak is a 2 × 2 matrix. The symbol for a block Toeplitz matrix is FN (z)= N−1 k ( )= ∞ k ε ∑k=0 Akz and the symbol for a block Toeplitz operator is F z ∑k=−∞ Akz . The pseudospectrum for a block Toeplitz matrix and a block Toeplitz operator are related by the following theorem from [2].
Theorem 3.3.1. Let A be an N × N Toeplitz matrix and let A be a Toeplitz operator. If
ε > 0, then
cl lim (Λε (AN )) = Λε (A), (3.1) N→∞ where cl is the closure operator.
The theorem implies that given a block triangular Toeplitz matrix, the ε-pseudospectrum is equal to the ε-pseudospectrum of a block triangular Toeplitz operator as the size of the matrix increases.
We consider the following symbol ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ ⎢11⎥ F(z)= ⎢ ⎥ ⎢ ⎥ ⎢z −1 ⎥ ⎣ ⎦ ⎡ ⎤ ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ √ √ ⎢11 ⎥ ⎢00 ⎥ with A = ⎢ ⎥ and A = ⎢ ⎥. The eigenvalues of F(z) are { 1 +z,− 1 +z}. Using 0 ⎢ ⎥ 1 ⎢ ⎥ ⎢0 −1 ⎥ ⎢10⎥ ⎣ ⎦ ⎣ ⎦ condition 1 of definition 2.1.2 and MATLAB, we plot the symbol and we test different sizes of ε and for the size of the matrix we let N = 50 and N = 100. The eigenvalues should be 1 and -1 so by running the different cases for N and ε we will determine if the mode will occur at 1 and -1. The perturbation code is used to implement the following images. The code has similarities for a Toeplitz matrix with scalar entries except the input of the block matrices.
size = 2; %Number of rows in base matrix
R1 = [1 1; 0 −1]; %first block in row
49 R2 = [0 0; 1 0]; %second block in row
R3 = [0 0; 0 0]; %third block in row
R4 = [0 0; 0 0]; %fourth block in row
R5 = [0 0; 0 0]; %fifth block in row
R6 = [0 0; 0 0]; %sixth block in row
R7 = [0 0; 0 0]; %seventh block in row
R8 = [0 0; 0 0]; %eight block in row
R9 = [0 0; 0 0]; %ninth block in row
R10 = [0 0; 0 0]; %tenth block in row
R = [R1 R2 R3 R4 R5 R6 R7 R8 R9 R10]; %builds the first row block
C1=R1; %
C2 = [0 0; 0 0]; %second block in column
C3 = [0 0; 0 0]; %third block in column
C4 = [0 0; 0 0]; %fourth block in column
C5 = [0 0; 0 0]; %fifth block in column
C6 = [0 0; 0 0]; %sixth block in column
C7 = [0 0; 0 0]; %seventh block in column
C8 = [0 0; 0 0]; %eight block in column
C9 = [0 0; 0 0]; %ninth block in column
C10 = [0 0; 0 0]; %tenth block in column
C = [ C1 ; C2 ; C3 ; C4 ; C5 ; C6 ; C7 ; C8 ; C9 ; C10 ] ;
matNo = 10; %Number of matrices
M = zeros( size*matNo, size*matNo ) ; M(1: size ,:) = R;
M(: ,1: size ) = C;
50 for n=1:matNo − 1
M(size*n+1:size*n+size , size*n+1:size*matNo)=R(: ,1: size*matNo− size*n);
M(size*n+1:size*matNo, size*n+1:size*n+size)=C(1: size*matNo− size*n ,:); end
M; %Block Toeplitz matrix
d=20;
yy=[];
for i =1:30
b=0.02; %epsilon
G=randn (d,d)+1 j* randn(d,d); %random matrix
E=b*G./max(svd(G)); %rescaled to have operator norm b lambda=eig (M+E);
yy=[yy ; lambda ];
end
theta = 0:.00001:2* pi ;
t=exp ( j* theta ); z=− sqrt ( t +1);
y=sqrt ( t +1);
plot (yy, ’o’)
grid on
hold on
plot (z) , axis ([− 1.5 1.5 −1.5 1.5])
hold on
plot (y) , axis ([− 1.5 1.5 −1.5 1.5])
hold on
51 s=real (yy ); q=imag (yy ); k=[s ,q]; figure (2); hist3 (k );
The following is a table of figures for different N and ε values for symbol ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ ⎢11⎥ F(z)= ⎢ ⎥. ⎢ ⎥ ⎢z −1 ⎥ ⎣ ⎦
52 ε = 0.02
N = 50 N = 100
ε = 0.01
N = 50 N = 100
ε = 0.002
N = 50 N = 100
Figure 3.6: Approximation of Λε (A) for block Toeplitz matrix A using different N and ε values. 53 From figure 3.6 we can see that as N increases and ε decreases we get the pseudospec trum of the block Toeplitz matrix to equal the pseudospectrum of the block Toeplitz operator. Therefore for further studying, we will consider the triangular block Toeplitz operator and look at further examples of block Toeplitz operators.
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56