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 − pseudospectrum of Davies’ complex
harmonic oscillator (3.1). From outside in, the curves correspond to =
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 σ (A) for = 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 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

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.

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 > 0 be arbitrary. The − pseudospectrum σ (A) of A is the set of
z ∈ C such that
(z − A) −1 > −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:
σ (A) is the set of z ∈ C such that
z ∈ σ(A + E)
for some E ∈ CN ∗N with E < (2.2)(1)
In words, the − pseudospectrum is the set of numbers that are eigenvalues of some
perturbed matrix A + E with E < .

Each of the definitions above follow on to show that the pseudospectra associated with
various are nested sets,
σ 1(A) ⊆ σ 2(A), 0 < 1 < 2,
(2.3) (1)
And that the intersection of all the pseudospectra is the spectrum,
>0
σ (A) = σ(A)
(2.4) (1)
Now lets define a third description of the − pseudospectra:
σ (A) is the set of z ∈ C such that
(z − A)v <
for some v ∈ CN with v = 1.
(2.5) (1)
The number z, in all the definitions, is known as an − pseudoeigenvalue of A, and v
would be the matching − pseudoeigenvector. So, similar to the definition of spectra, the
− pseudospectrum is the set of − 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 ∈ 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 < and some
non zero v ∈ CN , accept this to be normalized, V = 1.Then (z − A)v = Ev < , as
required.
To show (1.5) =⇒ (1.1), assume (z − A)v = su for some v, u ∈ CN with v = u = 1
and s < . Then (z − A)−1u = s−1v, so (z − A)−1 ≥ s−1 > −1.
Lastly, to show (1.1) =⇒ (1.2), assume (z − A)−1 > −1. Then (z − A)−1u = s−1v and
therefore zv − Av = su for some v, u ∈ CN with v = u = 1 and s < .

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, σ (A) is the set of z ∈ C such that
smin(z − A) <
(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 Schrödinger operator for a harmonic potential
acting in L2(R). This Schrödinger operator that is not real but complex, and is defined by:
Au = −d2u
dx2 + 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 − pseudospectrum of Davies’ complex harmonic
oscillator (3.1). From outside in, the curves correspond to = 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 = du
dx
(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) =
d
x ez(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 , σ (A) is equal to the half-plane lying to the left of some
line Rez = c 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 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 σ (A)
for = 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 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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