project report(1)

Basic Spectra and Pseudospectra of matrices and operators.

Different definitions, their equivalence.

22009818

Department of Mathematics and Statistics

University of Reading

March 1, 2016

Abstract

In certain cases the use of eigenvalues to analyse non-normal matrices can be un-

reliable, whereby the results can vary greatly from estimates. This is especially true

when the corresponding eigenvector sets are not well conditioned in regards to the norm

of applied interest. We consider the case of Euclidean or 2-norm, where the matrices

or operators are non-normal, with eigenvectors that are not orthogonal. In these cases

pseudospectra can be used to give a precise and graphical substitute for studying ma-

trices and operators that are non-normal. Here, through literature research, we show

examples of how pseudospectra is used to study matrices and operators. Previous re-

search suggests that non normal matrices and operators can show to a variety of unusual

and hard to predict behaviours.

i

Part 3 Project (Project Report) 22009818

Contents

List of Figures ii

1 Introduction 1

2 Definitions of pseudospectra and their equivalence 2

3 Operator example 5

4 Matrix examples 9

5 Conclusion 13

6 Bibliography 14

List of Figures

1 Taken from (1). Spectrum and ε − pseudospectrum of Davies’ complex

harmonic oscillator (3.1). From outside in, the curves correspond to ε =

10−1, 10−2, ..., 10−8. The resolvent norm grows exponentially as z → ∞along rays in the complex plane satisfying 0 < θ < π/2. . . . . . . . . . . . 5

2 Taken from (1). Pseudospectra of the differentiation operator A of (3.2)-(3.3)

for an interval length of d=2. The solid lines are the right hand boundaries

of σε(A) for ε = 10−1, 10−2, ..., 10−8 (from right to left). The dashed line,

the imaginary axis, is the right-hand boundary of the numerical range. If d

were increased, the ε levels would decrease exponentially. . . . . . . . . . . . 7

3 Taken from (1). For dRez << 0, the function u(x) = ezx and w(x) =

ezx − edRez+ixImz are ’nearly eigenfuctions of A, although neither is near

any eigenfunction. Notably u satifies the eigenvalue equation u′(x) = zx,

but not the boundary condition; w satifies the boundary condition, but not

the eigenvalue equation. Here d = 2, z = −2. . . . . . . . . . . . . . . . . . 7

4 Taken from (2). Spectrum and epsilon-pseudospectra for the shift matrix

of dimension 10 (left) and dimension 100 (right). The small matrix has a

markedly high degree of non-normality, which blows up as the dimension is

increased. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

5 Taken from (2). Grcar matrix of dimension N=100. The map of the symbol,

a(T ) (left) and the spectrum and epsilon-pseudospectra (right). . . . . . . . 11

6 Taken from (2). Daisy matrix of dimension N=200. The map of the symbol,

a(T ) (left) and the spectrum and epsilon-pseudospectra (right). . . . . . . . 11

MA3PR ii Professor Michael Levitin


7 Taken from (2). Truncation of a Toeplitz operator with a hole in the spec-

trum. The map of the symbol, a(T ) (left) and the spectrum and epsilon-

pseudospectra for a matrix of dimension N=100(right). . . . . . . . . . . . . 12

MA3PR iii Professor Michael Levitin


1 Introduction

Past study and analysis of linear models has been carried out using eigenvalues. This

has been a very successful tool for solving many problems throughout mathematics. It is

particularly good for problems involving self- adjoint matrices and operators, which possess a

basis of orthogonal eigenvectors. Fields in which the application of eigenvalue techniques has

been a success include quantum mechanics, fluid mechanics, functional analysis, structural

analysis, numerical analysis and acoustics.

However the use of eigenvalues is not as good once there are problems that include

matricies or operators that lack an orthogonal basis of eigenvectors. These operators are

defined as non-normal. This is a property which can lead to a range of behaviours. As an

example, transient behaviour might be connected with non-normality, which is completely

changed from the asymptotic behaviour suggested by eigenvalues. Non-normality is

important in areas such as meteorology, control theory, matrix iteration and analysis of

high-powered lasers.

Various techniques have been put into practice to analyse and describe non-normality.

Such examples are: numerical range, angles between invariant subspaces, and the

condition numbers of eigenvalues. Nevertheless, pseudospectra offers a different graphical

and analytical method for studying non-normal matrices and operators. Therefore, that is

what we will focus on in this report.

Definition of eigenvalues

Let A ∈ C be a N by N matrix.

Let v ∈ C be a non-zero column vector of length N .

Let λ ∈ C be scalar.

Then v is an eigenvector of A, and λ is the eigenvalue, if Av = λv.

The set of all eigenvalues of A is the spectrum of A. The spectrum defines a non-empty

subset of the complex plane. The spectrum is denoted by σ(A). The spectrum is also

defined as the set of points z ∈ C where the resolvent matrix, (z −A)−1, does not exit.

This paragraph follows (2). The eigenvalues described in the previous paragraph where

just for matrices. If we take A to be a linear operator such as a differential, an integral

operator or an infinite matrix. The spectrum, σ(A), is then defined in a Banach or Hilbert

space. This is the set of numbers z ∈ C for which the resolvent (z −A)−1 does not exist as

a bounded operator defined on the whole space. One of the significances of this is that not

all z ∈ σ(A) has to be an eigenvalue. Banach and Hilbert spaces will be discussed further

in the main text of this report.

MA3PR 1 Professor Michael Levitin


2 Definitions of pseudospectra and their equivalence

Now that we have discussed why we want to study pseudospectra. We begin the main text

of this report by looking at the definitions of pseudospectra and their resulting equivalence.

In this section we follow (1). For the following definitions we will consider eigenvalues in

matrices. The concept of pseudospectra is driven from the question is ‖A−1‖ large?

Though, the condition that defines eigenvalues is a condition of matrix singularity. To say

z ∈ C is an eigenvalue of A, is the same thing as to say z −A is singular. However, what

defines being an eigenvalue of a matrix is not concrete. Consequently an improved question

would be is ‖(z −A)‖−1 large? This takes us to the first definition of pseudospectra:

Let A ∈ CN ∗N and ε > 0 be arbitrary. The ε− pseudospectrum σε(A) of A is the set of

z ∈ C such that

‖(z −A)‖−1 > ε−1

(2.1) (1)

The matrix (z −A)−1 is known as the resolvent of A at z. !!!!

It is intuitive to think that (z −A)−1 is large when z is close to the eigenvalue of A. This

may lead to wondering about the importance of psuedospectra. However pseudospectra

becomes significant for matrices far from normal. Where (z −A)−1 may be large whilst z

is far from the spectrum. The second definition of pseudospectra comes from the

connection amongst the resolvent norm and eigenvalue perturbation theory:

σε(A) is the set of z ∈ C such that

z ∈ σ(A+ E)

for some E ∈ CN ∗N with ‖E‖ < ε (2.2)(1)

In words, the ε− pseudospectrum is the set of numbers that are eigenvalues of some

perturbed matrix A+ E with‖E‖ < ε.



Each of the definitions above follow on to show that the pseudospectra associated with

various ε are nested sets,

σε1(A) ⊆ σε2(A), 0 < ε1 < ε2,

(2.3) (1)

And that the intersection of all the pseudospectra is the spectrum,

⋂ε>0

σε(A) = σ(A)

(2.4) (1)

Now lets define a third description of the ε− pseudospectra:

σε(A) is the set of z ∈ C such that

‖(z −A)v‖ < ε

for some v ∈ CN with ‖v‖ = 1.

(2.5) (1)

The number z, in all the definitions, is known as an ε− pseudoeigenvalue of A, and v

would be the matching ε− pseudoeigenvector. So, similar to the definition of spectra, the

ε− pseudospectrum is the set of ε− pseudoeigenvalues.

We form the equivalence of the definitions of pseudospectra by the theorem:

For any matrix A ∈ CN ∗N , the three definitions above are equivalent. (2.6)(1)

Proof: For this proof we follow (1). In the case where z ∈ σ(A) the equivalence is trivial.

Now we look at z 6∈ σ(A), which suggests that (z −A)−1 exists.

To show (1.2) =⇒ (1.5), let (A+ E)v = zv for some E ∈ CN ∗N with ‖E‖ < ε and some

non zero v ∈ CN , accept this to be normalized, ‖V ‖ = 1.Then ‖(z −A)v‖ = ‖Ev‖ < ε, as

required.

To show (1.5) =⇒ (1.1), assume (z −A)v = su for some v, u ∈ CN with ‖v‖ = ‖u‖ = 1

and s < ε. Then (z −A)−1u = s−1v, so ‖(z −A)−1‖ ≥ s−1 > ε−1.

Lastly, to show (1.1) =⇒ (1.2), assume ‖(z −A)−1‖ > ε−1. Then (z −A)−1u = s−1v and

therefore zv −Av = su for some v, u ∈ CN with ‖v‖ = ‖u‖ = 1 and s < ε.



Until now we let ‖ · ‖ be an arbitrary norm. Once we use linear operators later in the

report, this will correspond to a setting of Banach spaces. Nevertheless next we look at

the additional properties that arise in Hilbert spaces. lets restrict focus to the instance

where CN is given the standard inner product:

(u, v) = u ∗ V

(2.7) (1)

and ‖ · ‖ is the corresponding 2-norm,

‖v‖ = ‖v‖2 =√v ∗ v

(2.8) (1)

These inner product and norm mean the Hermitian conjugate of a matrix is the same as

its adjoint, for which the symbol A∗ is used. In the case where more general inner

products and norms are wanted, we can deal with within the context of the 2-norm. This

is done by doing a similarity transformation A→ DAD−1, where D is nonsingular.

If ‖ · ‖ = ‖ · ‖2, the norm of a matrix is its largest singular value and the norm of the

inverse is the inverse of the smallest singular value. Particularly,

‖(z −A)−1‖2 = [smin(z −A)]−1

(2.9) (1)

where smin(z −A) represents the smallest singular value of z −A. This proposes a fourth

definition of pseudospectra:

For ‖ · ‖ = ‖ · ‖2, σε(A) is the set of z ∈ C such that

smin(z −A) < ε

(2.10)(1)

From (2.9) it is apparent that (2.10) is equivalent to (2.1), so likewise it must be

equivalent to the definitions (2.2) and (2.5) of pseudospectra.



3 Operator example

Throught this chapter we follow (1). The ensuing simple operator example will

demonstrate the spectacular occurrence of pseudospectra and how it deviates strongly

from normality. However, first I will present a highly non-normal operator shown by E.B

Davies in (3) and (4). It is an example of a Schrodinger operator for a harmonic potential

acting in L2(R). This Schrodinger operator that is not real but complex, and is defined by:

Au = −d2udx2

+ ix2u, x ∈ R

(3.1) (1)

Davies showed how this operator deviates strongly from normality, which is demostrated

in figure 3.1 bellow.

Figure 1: Taken from (1). Spectrum and ε− pseudospectrum of Davies’ complex harmonic

oscillator (3.1). From outside in, the curves correspond to ε = 10−1, 10−2, ..., 10−8. The

resolvent norm grows exponentially as z → ∞ along rays in the complex plane satisfying

0 < θ < π/2.



However, for the example in this report we will look at the first derivative operator, as

shown bellow, which is a much less complex example.

Au = u′ = dudx

(3.2)

in the space L2(0, d), subject to the boundary condition:

u(d) = 0

(3.3)

The eigenfunctions of A require the form ezx, however no functions of this form satisfy the

boundary condition. Therefore there are no eigenfunctions, and thus the empty set

σ(A) = θ is the spectrum of A. This is proved by showing that the resolvent (z −A)−1

exists as a bounded operator for any z ∈ C, the equation for this is:

(z −A)−1v(x) =∫ dx e

z(x−s)v(s)ds

(3.4)(1)

By using the method of variation of parameters the equation can be derived and applied

to the ordinary differential equation zu− u′ = v.



On the other hand, we will now look at the pseudospectra of A. From (3.4) we can see

that while the resolvent norm ‖(z −A)−1‖ is finite for every z, it is huge when z is far

inside the left half-plane, increasing exponentially as a function of exp(−dRez). (3.4) also

shows how ‖(z −A)−1‖ relies on Rez but not Imz. To prove this, we show that for any

z ∈ C, v(s) and α ∈ R, the pairs z, v(s), and z + iα, eiαsv(s) lead to the same norm of the

integral in (3.4). Hence, each ε, σε(A) is equal to the half-plane lying to the left of some

line Rez = cε in the complex plane. Figure 3.2 bellow shows this, where the stand out part

to take notice of is the very quick decrease of ε as one moves into the left half-plane.

Figure 2: Taken from (1). Pseudospectra of the differentiation operator A of (3.2)-(3.3)

for an interval length of d=2. The solid lines are the right hand boundaries of σε(A)

for ε = 10−1, 10−2, ..., 10−8 (from right to left). The dashed line, the imaginary axis, is the

right-hand boundary of the numerical range. If d were increased, the ε levels would decrease

exponentially.

Following this, the obvious question is, why does A have a massive resolvent norm in the

left half-plane? An explination is put forward by figure 3.3 bellow.

Figure 3: Taken from (1). For dRez << 0, the function u(x) = ezx and w(x) = ezx −edRez+ixImz are ’nearly eigenfuctions of A, although neither is near any eigenfunction.

Notably u satifies the eigenvalue equation u′(x) = zx, but not the boundary condition; w

satifies the boundary condition, but not the eigenvalue equation. Here d = 2, z = −2.



The function

u(x) = ezx, z ∈ C

(3.5) (1)

does not satify (3.3), but it nearly does for Rez << 0. Therefore u is not an eigenfunction

of A or near to any eigenfunction. But if you look at it as an eigenfunction of a slightly

perturbed problem, then it could be described as ’nearly an eigenfunction’. Still, it can

not be called a pseudoeigenfunction, or pseudomode, as u does not go to the domain of

the operator, due to the fact it infinges the boundary condition. One way around this is to

adapt u(x) by subtraction of a small term like edRez+ixImz, so it becomes a proper

pseudomode. Then a lower bound for ‖(z −A)−1‖ can be derived. Satish Reddy showed

you can also determine ‖(z −A)−1‖ exactly by calculus of variations, however the result is

not a closed formula.



4 Matrix examples

The matrix examples in this chapter may be slightly disjoint from the report, however

they are fascinating and provide not only a different, but a visually striking look at

pseudospectra. Toeplitz matrices supply One of the most interesting uses of

pseudospectra. In a Toeplitz matrix, each descending diagonal from left to right is

constant. The corresponding Toeplitz operator is a singly-infinite matrix. The constants

on the diagonals are the Laurent coefficients denoted by the symbol, T . The symbol, a, is

used to determine the spectrum of a Toeplitz operator.

Many fields such as the finite difference discretizations of differential equations include

banded Toeplitz matrices. An instance of where it would be useful to know the

eigenvalues of this matrix is to determine whether the discretization of an initial boundary

value problem is stable.

Suppose the Toeplitz matrix T is defined entry wise,

Tjk = tJ−k

.

The coefficients tj define a function a by the Laurent series

a(eiθ) =∑∞

j=−∞ tjeijθ, θ ∈ [0, 2π)

(4.1) (2)

From (2) we know that if the Toeplitz matrix T is banded, the there exist some j0 such

that tj = 0 whenever ‖J‖ > j0. This implies that the function a is a trigonometric

polynomial.

Let TN be used to denote the N-by-N finite section of the infinite dimensional Toeplitz

matrix T .

The eigenvalues of finite banded Toeplitz matrices lie on curves in the complex plane that

Schmidt and Spitzer have characterized. Our argument now follows (2). Contrastingly, the

spectrum of the corresponding infinite dimensional Toeplitz operator is the set of points in

the complex plane that a(T ) encloses with non-zero winding number. A winding number

of a closed curve in the plane around a given point, is an integer representing the total

number of times that curve travels counterclockwise around the point. Therefore, the limit

of the spectrum of banded Toeplitz matrices as the dimension goes to infinity is usually

highly altered from the spectrum of the limit.



The resolution for this situation is done by looking at the behaviour of pseudospectra.

Although, the eigenvalues of the finite Toeplitz matrices might lie on curves in the

complex plane. It has been detected by Landau, Reichel, Trefethen and Bttcher that the

resolvent norm grows exponentially in the matrix dimension for all z in the interior of the

spectrum corresponding to the infinite dimensional operator. The outcome of this means

its difficult to precisely calculate eigenvalues of non-symmetric banded Toeplitz matrices.

This is true even for matrices of comparatively modest dimensions. This causes caution to

basing analysis of finite Toeplitz matrices based on eigenvalues alone. Additionally, it has

been established by the same researchers as above that the pseudospectra of TN converge

to the pseudospectra of T .

Next we will look at examples of computed pseudo spectra of finite Toeplitz matrices,

found on (2). The distance between the pseudospectral boundary and the eigenvalues even

for small values of epsilon is a reoccurring theme throughout these computations.

The first example will be the most simple non-symetric Toeplitz matrix, the shift operator

a(t) = t. The infinite dimensional version is a classic example in operator theory. Also

finite shift operators occur in linear algebra in the Jordan cannonical form.

Figure 4: Taken from (2). Spectrum and epsilon-pseudospectra for the shift matrix of

dimension 10 (left) and dimension 100 (right). The small matrix has a markedly high

degree of non-normality, which blows up as the dimension is increased.



A different well known example is the Grcar matrix, which was first documented by L. N.

Trefethen. It is determined by the symbol a(t) = −t+ 1 + t−1 +t−2 + t−3.

Figure 5: Taken from (2). Grcar matrix of dimension N=100. The map of the symbol, a(T )

(left) and the spectrum and epsilon-pseudospectra (right).

Our next examples pseudospectra looks like petals of a flower, this is shown in Figure 3.

The symbol that corresponds to this is a(t) = −t+ t−5.

Figure 6: Taken from (2). Daisy matrix of dimension N=200. The map of the symbol, a(T )

(left) and the spectrum and epsilon-pseudospectra (right).



The last example, Figure 4, is from a banded Toeplitz matrix with a hole in the spectrum

of the corresponding infinite dimensional operator. It is notable how the pseudospectra do

not grow quickly in the interior region where the winding number of a(T ) is zero.

Figure 7: Taken from (2). Truncation of a Toeplitz operator with a hole in the spectrum.

The map of the symbol, a(T ) (left) and the spectrum and epsilon-pseudospectra for a matrix

of dimension N=100(right).



5 Conclusion

The examples put in this report, especially the matrix ones, were aimed to show the

geometrical feature of non-normality. Whilst simultaneously persuading the person who

reads it of its beauty. Also the report intended to present how pseudospectra can be both

informative and applicable. In conclusion both matrices and operators can be captivating

when studying the effects of non-normality. Problems involving eigenvalues occur in many

aspects of mathematics, most frequently in the form of differential or integral operators

which may be reduced to huge matrices. The majority of these matrices will be

non-symmetric, and thus in most cases non-normal. Pseudospectra provides a way of

exploring these and informing us about the eigenvalues.



6 Bibliography

[1] Llyod N. Trefethen and Mark Embree (2005) SPECTRA AND PSEUDOSPECTRA

The Behavior of Nonnormal Matrices and Operators, Princeton University Press

[2] Mark Embree and Lloyd N. Trefethen. Pseudospectra Gateway. Web site:

http://www.comlab.ox.ac.uk/pseudospectra

[3] E. B. Davies. Pseudo-spectra, the harmonic oscillator and complex resonances. Proc.

Roy. Soc. Lond. Ser. A 455 (1999), 585-599.

[4] E. B. Davies. Semi-classical states for non-self-adjoint Schrodinger operators.

Commun. Math. Phys. 200 (1999), 35-41.


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