EXPLICIT BOUNDS FOR THE PSEUDOSPECTRA OF VARIOUS CLASSES...

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EXPLICIT BOUNDS FOR THE PSEUDOSPECTRA OF VARIOUS CLASSES OF MATRICES AND OPERATORS FEIXUE GONG 1 , OLIVIA MEYERSON 2 , JEREMY MEZA 3 , ABIGAIL WARD 4 MIHAI STOICIU 5 (ADVISOR) SMALL 2014 - MATHEMATICAL PHYSICS GROUP Abstract. We study the ε-pseudospectra σε(A) of square matrices A C N×N . We give a complete characterization of the ε-pseudospectrum of any 2×2 matrix and describe the asymptotic behavior (as ε 0) of σε(A) for any square matrix A. We also present explicit upper and lower bounds for the ε-pseudospectra of bidiagonal and tridiagonal matrices, as well as for finite rank operators. Contents 1. Introduction 2 2. Pseudospectra 2 2.1. Motivation and Definitions 2 2.2. Normal Matrices 6 2.3. Non-normal Diagonalizable Matrices 7 2.4. Non-diagonalizable Matrices 8 3. Pseudospectra of 2 × 2 Matrices 10 3.1. Non-diagonalizable 2 × 2 Matrices 10 3.2. Diagonalizable 2 × 2 Matrices 12 4. Asymptotic Union of Disks Theorem 15 5. Pseudospectra of N × N Jordan Block 20 6. Pseudospectra of bidiagonal matrices 22 6.1. Asymptotic Bounds for Periodic Bidiagonal Matrices 22 6.2. Examples of Bidiagonal Matrices 29 7. Tridiagonal Matrices 32 Date : October 30, 2014. 1,2,5 Williams College 3 Carnegie Mellon University 4 University of Chicago. 1

Transcript of EXPLICIT BOUNDS FOR THE PSEUDOSPECTRA OF VARIOUS CLASSES...

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EXPLICIT BOUNDS FOR THE PSEUDOSPECTRA OF VARIOUS

CLASSES OF MATRICES AND OPERATORS

FEIXUE GONG1, OLIVIA MEYERSON2, JEREMY MEZA3, ABIGAIL WARD4

MIHAI STOICIU5 (ADVISOR)

SMALL 2014 - MATHEMATICAL PHYSICS GROUP

Abstract. We study the ε-pseudospectra σε(A) of square matricesA ∈ CN×N .

We give a complete characterization of the ε-pseudospectrum of any 2×2 matrixand describe the asymptotic behavior (as ε→ 0) of σε(A) for any square matrix

A. We also present explicit upper and lower bounds for the ε-pseudospectra

of bidiagonal and tridiagonal matrices, as well as for finite rank operators.

Contents

1. Introduction 2

2. Pseudospectra 2

2.1. Motivation and Definitions 2

2.2. Normal Matrices 6

2.3. Non-normal Diagonalizable Matrices 7

2.4. Non-diagonalizable Matrices 8

3. Pseudospectra of 2 × 2 Matrices 10

3.1. Non-diagonalizable 2 × 2 Matrices 10

3.2. Diagonalizable 2 × 2 Matrices 12

4. Asymptotic Union of Disks Theorem 15

5. Pseudospectra of N ×N Jordan Block 20

6. Pseudospectra of bidiagonal matrices 22

6.1. Asymptotic Bounds for Periodic Bidiagonal Matrices 22

6.2. Examples of Bidiagonal Matrices 29

7. Tridiagonal Matrices 32

Date: October 30, 2014.1,2,5Williams College 3Carnegie Mellon University 4University of Chicago.

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8. Finite Rank operators 33

References 35

1. Introduction

The pseudospectra of matrices and operators is an important mathematical objectthat has found applications in various areas of mathematics: linear algebra, func-tional analysis, numerical analysis, and differential equations. An overview of themain results on pseudospectra can be found in [14].

In this paper we investigate a few classes of matrices and operators and provideexplicit bounds on their ε-pseudospectra. We study 2 × 2 matrices, bidiagonaland tridiagonal matrices, as well as finite rank operators. We also describe theasymptotic behavior of the ε-pseudospectrum σε(A) of any n × n matrix A.

The paper is organized as follows: in Section 2 we give the three standard equivalentdefinitions for the pseudospectrum and we present the “classical” results on ε-pseudospectra of normal and diagonalizable matrices (the Bauer-Fike theorems).Section 3 contains a detailed analysis of the ε-pseudospectrum of 2 × 2 matrices,including both the non-diagonalizable case (Subsection 3.1) and the diagonalizablecase (Subsection 3.2). The asymptotic behavior (as ε→ 0) of the ε-pseudospectrumof any n×n matrix is described in Section 4, where we show (in Theorem 4.1) that,for any square matrix A, the ε-pseudospectrum converges, as ε → 0 to a union ofdisks.

The main result of Section 4 is applied to several classes of matrices: Jordan blocks(Section 5), bidiagonal matrices (Section 6), and tridiagonal matrices (Section 7).In Section 5 we derive explicit lower bounds for σε(J), where J is any Jordan block,while Section 6 is dedicated to the analysis of arbitrary periodic bidiagonal matricesA. We derive explicit formulas (in terms the coefficients of A) for the asymptoticradii of the ε-pseudospectrum of A, as ε→ 0. We continue with a brief investigationof tridiagonal matrices in Section 7, while in the last section (Section 8) we considerfinite rank operators and show that the ε-pseudospectrum of an operator of rank

m is at most as big as Cε1m , as ε→ 0.

2. Pseudospectra

2.1. Motivation and Definitions. The concept of the spectrum of a matrix A ∈CN×N provides a fundamental tool for understanding the behavior of A. As is well-known, a complex number z ∈ C is in the spectrum of A (denoted σ(A)) wheneverzI − A (which we will denote as z − A) is not invertible, i.e., the characteristicpolynomial of A has z as a root. As slightly perturbing the coefficients of A willchange the roots of the characteristic polynomial, the property of “membershipin the set of eigenvalues” is not well-suited for many purposes, especially those innumerical analysis. We thus want to find a characterization of when a complex

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EXPLICIT BOUNDS FOR THE PSEUDOSPECTRA OF MATRICES AND OPERATORS 3

number is close to an eigenvalue, and we do this by considering the set of complexnumbers z such that ∥(z−A)−1∥ is large, where the norm here is the usual operatornorm induced by the Euclidean norm, i.e.

∥A∥ = sup∥v∥=1

∥Av∥.

The motivation for considering this question comes from the following observation:

Proposition 2.1. Let A ∈ CN×N , λ an eigenvalue of A, and zn be a sequence ofcomplex numbers converging to λ with zn ≠ z for all n. Then ∥(zn −A)−1∥ →∞ asn→∞.

Proof. Let v be the eigenvector corresponding to λ with norm 1, so Av = λv. Define

vn =(zn −A)v

∥(zn −A)v∥(note that (zn −A)v ≠ 0 by assumption). Now, note that

∥(zn −A)−1∥ ≥ ∥(zn −A)−1vn∥

= ∥(zn −A)−1(zn −A)v∥(zn −A)v∥

= ∥ v

∥(zn −A)v∥∥

= 1

∣λ − zn∣,

and since the last quantity gets arbitrarily large for large n, we obtain the result. �

We call the operator (z −A)−1 the resolvent of A. The observation that the normof the resolvent is large when z is close to an eigenvalue of A leads us to the firstdefinition of the ε-pseudospectrum of an operator.

Definition 2.1. Let A ∈ CN×N , and let ε > 0. The ε-pseudospectrum of A is theset of z ∈ C such that

∥(z −A)−1∥ > 1/ε

Note that the boundary of the ε-pseudospectrum is exactly the 1/ε level curve ofthe function z ↦ ∥(z − A)−1∥. Fig. 2.1 depicts the behavior of this function nearthe eigenvalues, as described in Prop. 2.1.

The resolvent norm has singularities in the complex plane, and as we approach thesepoints, the resolvent norm grows to infinity. Conversely, if ∥(z −A)−1∥ approachesinfinity, then z must approach some eigenvalue of A [14, Thm 2.4].

(It is also possible to develop a theory of pseudo-spectrum for operators on Banachspaces, and it is important to note that this converse does not necessarily hold forsuch operators; that is, there are operators [4, 5] such that ∥(z −A)−1∥ approachesinfinity, but z does not approach the spectrum of A.)

The second definition of the ε-pseudospectrum arises from eigenvalue perturbationtheory [8].

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4 GONG, MEYERSON, MEZA, WARD, STOICIU

Figure 2.1. Contour Plot of Resolvent Norm

Definition 2.2. Let A ∈ CN×N . The ε-pseudospectrum of A is the set of z ∈ Csuch that

z ∈ σ(A +E)for some E with ∥E∥ < ε.

Finally, the third definition of the ε-pseudospectum is the following:

Definition 2.3. Let A ∈ CN×N . The ε-pseudospectrum of A is the set of z ∈ Csuch that

∥(z −A)v∥ < εfor some unit vector v.

This definition is similar to our first definition in that it quantifies how close zis to an eigenvalue of A. In addition to this, it also gives us the notion of anε-pseudoeigenvector.

Theorem 2.1 (Equivalence of the definitions of pseudospectra). For any matrixA ∈ CN×N , the three definitions above are equivalent.

The proof of this theorem follows [14, §2].

Proof. Define the following sets

Mε = {z ∈ C ∣ ∥(z −A)−1∥ > ε−1}

Nε = {z ∈ C ∣ z ∈ σ(A +E) for some E ∈ CN×N , ∥E∥ < ε}Pε = {z ∈ C ∣ ∥(z −A)v∥ < ε for some v ∈ CN , ∥v∥ = 1}

To show equivalence, we will show that Mε ⊆ Nε ⊆ Pε ⊆Mε.

Nε ⊆ Pε

Let z ∈ Nε. Let E be such that z ∈ σ(A+E), with ∥E∥ < ε. Let v be an eigenvectorof norm one for z, that is ∥v∥ = 1 and (A +E)v = zv. Then

∥(z −A)v∥ = ∥(z −A −E)v +Ev∥ ≤ ∥E∥∥v∥ < εand hence z ∈ Pε.

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EXPLICIT BOUNDS FOR THE PSEUDOSPECTRA OF MATRICES AND OPERATORS 5

Pε ⊆Mε

Let z ∈ Pε. Let v ∈ CN be such that ∥v∥ = 1 and ∥(z −A)v∥ < ε. Observe

(z −A)v = ∥(z −A)v∥ (z −A)v∥(z −A)v∥

= su

where we define u = (z−A)v∥(z−A)v∥

and s = ∥(z−A)v∥. Then, we see that s−1v = (z−A)−1u,

which implies that

∥(z −A)−1∥ ≥ ∥(z −A)−1u∥ = ∣s−1∣∥v∥ = s−1 > ε−1

and hence z ∈Mε.

Mε ⊆ Nε

Let z ∈ Mε. We will construct an E with ∥E∥ < ε and z ∈ σ(A + E). Since∥(z −A)−1∥ = sup

∥v∥=1 ∥(z −A)−1v∥ > ε−1, there exists some v with ∥v∥ = 1 such that

∥(z −A)−1v∥ > ε−1. Observe

(z −A)−1v = ∥(z −A)−1v∥ (z −A)−1v∥(z −A)−1v∥

= ∥(z −A)−1v∥u

where u is the unit vector (z−A)−1v

∥(z−A)−1v∥. So v

∥(z−A)−1v∥= (z −A)u, from which we have

zu = Au + v

∥(z −A)−1v∥

Define E = v∥(z−A)−1v∥

u∗. Note Eu = v∥(z−A)−1v∥

and ∥E∥ < ε. Then we have zu =Au +Eu, and so z ∈ σ(A +E). �

As all three definitions are now shown to be equivalent, we can unambiguouslydenote the ε-pseudospectrum of A as σε(A).

Fig. 2.2 depicts an example of ε-pseudospectra for a specific matrix and for var-ious ε. We see that the boundaries of ε-pseudospectra for a matrix are curvesin the complex plane around the eigenvalues of the matrix. We are interested inunderstanding geometric and algebraic properties of these curves.

Figure 2.2. The curves bounding the ε-pseudospectra of an op-erator A, for different values of ε.

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6 GONG, MEYERSON, MEZA, WARD, STOICIU

When a matrix A is the direct sum of smaller matrices, we can look at the pseu-dospectra of the smaller matrices to understand the ε-pseudospectrum of A. Weget the following theorem from [14]:

Theorem 2.2.

σε(A1 ⊕A2) = σε(A1) ∪ σε(A2).

2.2. Normal Matrices. Recall that a matrix A is normal if AA∗ = A∗A, or equiv-alently if A can be diagonalized with an orthonormal basis of eigenvectors.

The pseudospectra of these matrices is particularly well-behaved: Thm. 2.3 showsthat the ε-pseudospectrum of a normal matrix is exactly a disk of radius ε aroundeach eigenvalue, as in shown in Fig. 2.3. This is clear for diagonal matrices; itfollows for normal matrices since as we shall see, the ε-pseudospectrum of a matrixis invariant under a unitary change of basis.

Figure 2.3. The ε-pseudospectrum of a normal matrix. Notethat each boundary is a perfect disk around an eigenvalue.

Theorem 2.3. Let A ∈ CN×N . Then,

(2.1) σ(A) +B(0, ε) ⊆ σε(A) for all ε > 0

and if A is normal and the operator norm is the 2-norm, then

(2.2) σε(A) = σ(A) +B(0, ε) for all ε > 0

Conversely, (2.2) implies that A is normal.

Proof. Let z ∈ σ(A) +B(0, ε). Write z = λ + w for some λ ∈ σ(A) and w ∈ C with∣w∣ < ε. Let v be an eigenvector for A with eigenvalue λ. Then, v is an eigenvectorfor the perturbed system A+wI with eigenvalue z. Thus, z ∈ σ(A+wI), and hencez ∈ σε(A). This shows (2.1).

To show (2.2), we now suppose A is a normal matrix. It suffices to show thebackwards inclusion to (2.1). Write A = UΛU∗ for a unitary U and diagonal Λ. Wehave

∥(z −A)−1∥ = ∥(UzU∗ −UΛU∗)−1∥ = ∥U(z −Λ)−1U∗∥ = ∥(z −Λ)−1∥ = 1

dist(z, σ(A))

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EXPLICIT BOUNDS FOR THE PSEUDOSPECTRA OF MATRICES AND OPERATORS 7

which implies that if z ∈ σε(A), then dist(z, σ(A)) < ε. Rewriting this, we getσε(A) ⊆ σ(A) +B(0, ε).

The converse of this proof follows from the discussion below of the condition numberand the full proof is postponed for now; see Corollary 2.1.

2.3. Non-normal Diagonalizable Matrices. Now suppose A is diagonalizablebut not normal, i.e. we cannot diagonalize A by an isometry of CN . In thiscase we do not expect to get an exact characterization of the ε-pseudospectra aswe did previously. The following example shows that there exist matrices withpseudospectra that are larger than the disk of radius ε:

Example 2.1. For the non-normal diagonalizable matrix A = (1 −10 2

),

B(2,Cε) ⊆ σε(A)

for any C <√

2 as ε→ 0.

Proof. Let z ∈ B(2,Cε), where C <√

2. We will construct E with ∥E∥ < ε so thatz is an eigenvalue of A +E.

Let E = ( 0 0−αε αε

) where ∣α∣ < 1√

2. Direct calculation gives ∥E∥ =

√2∣α∣ε < ε. We

find the eigenvalues of A +E:

det(A +E − zI) = (1 − z)(2 + αε − z) − αε = z2 − z(3 + αε) + 2 = 0

The roots of this equation are given by

z = 3 + αε ±√

1 + 6αε + α2ε2

2.

Recall for x small, the Taylor expansion (1 + x)p = 1 + px +O(x2). This gives thatthe root in which we add the discriminant has the expansion

z = 3 + αε + (1 + 3αε +O(ε2))2

Ô⇒ z = 2 + 2αε +O(ε2)

Since we may take any α with ∣α∣ < 1√

2, we obtain the result.

We begin here the characterization of the behavior of non-normal, diagonalizablematrices. Write A = V DV −1. Define the condition number of V as

κ(V ) = ∥V ∥∥V −1∥ = smax(V )smin(V )

where smax(V ) and smin(V ) are the maximum and minimum singular values of V ,respectively. Note that there is some ambiguity when we define κ(V ), as V is notuniquely determined. If the eigenvalues are distinct, then κ(V ) becomes unique ifthe eigenvectors are normalized by ∥vj∥ = 1.While this choice may not necessarily

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8 GONG, MEYERSON, MEZA, WARD, STOICIU

be the one that minimizes κ(V ), [13] shows that this choice exceeds the minimal

value by at most a factor of√N .

Note that κ(V ) is necessarily greater than or equal to 1, and we claim κ(V ) = 1if and only if A is normal. The backward direction follows from our unitarilyequivalent norm: if A is normal, then V is unitary, which implies that ∥V ∥ = ∥V −1∥ =1 and hence κ(V ) = 1. To show the forward direction, suppose κ(V ) = smax(V )

smin(V )= 1.

Thus, in the singular value decomposition of V , the singular values are all the same,which implies that V = sUW ∗ for s > 0 and unitary matrices U,W . Define V ′ = V /s,and note that V ′ is a unitary matrix. Also note

V ′DV ′−1 = (V /s)D(sV −1) = V DV −1 = Aand thus A is unitarily diagonalizable and hence normal.

The condition number gives an upper bound on the size of the pseudo spectrum ofa non-normal diagonalizable matrix:

Theorem 2.4 (Bauer-Fike). Let A ∈ CN×N and let A be diagonalizable, A =V DV −1. Then for each ε > 0,

σ(A) +B(0, ε) ⊆ σε(A) ⊆ σ(A) +B(0, εκ(V ))where

κ(V ) = ∥V ∥∥V −1∥ = smax(V )smin(V )

.

Corollary 2.1. If σε(A) = σ(A) +B(0, ε), then A is normal.

Proof (Thm. 2.4). The first inclusion was established in the previous theorem. Forthe second inclusion, let z ∈ σε(A). Then

(z −A)−1 = (z − V DV −1)−1 = [V (z −D)V −1]−1 = V (z −D)−1V −1

which implies that

ε−1 < ∥(z −A)−1∥ ≤ κ(V )∥(z −D)−1∥ = κ(V )dist(z, σ(A))

,

whence

dist(z, σ(A)) < εκ(V ).�

2.4. Non-diagonalizable Matrices. So far we have considered normal matrices,and more generally diagonalizable matrices. We now relax our constraint that ourmatrix be diagonalizable, and provide similar bounds on the pseudospectra. Whilenot every matrix is diagonalizable, every matrix can be put in Jordan normal form.Below we give a brief review of the Jordan form.

Let A ∈ CN×N and suppose A has only one eigenvalue, λ with geometric multiplicityone. Writing A in Jordan form, there exists a matrix V such that AV = V J , whereJ is a single Jordan block of size N . Write

V = (v1, v2, . . . , vn)

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EXPLICIT BOUNDS FOR THE PSEUDOSPECTRA OF MATRICES AND OPERATORS 9

Then,

AV = (Av1, Av2, . . . , Avn) = (λv1, v1 + λv2, . . . , vn−1 + λvn) = V J,

where v1 is the right eigenvector associated with λ and v2, ..., vn are generalizedright eigenvectors, that is right eigenvectors for (A − λI)k for k > 1.

We also know that there exists a matrix U∗ = V −1 such that U∗A = JU∗.

U =⎛⎜⎜⎜⎝

u1u2⋮un

⎞⎟⎟⎟⎠.

Then,

U∗A =⎛⎜⎜⎜⎝

u1Au2A⋮

unA

⎞⎟⎟⎟⎠=

⎛⎜⎜⎜⎜⎜⎝

λu1 + u2λu2 + u3

⋮λun−1 + un

λun

⎞⎟⎟⎟⎟⎟⎠

= JU∗.

Thus, un is the left eigenvector associated with λ and u1, ..., un−1 are generalizedleft eigenvectors.

Similar to how κ(V ) quantifies the normality of a matrix, we can also quantify thenormality of an eigenvalue:

Definition 2.4. For any simple eigenvalue λj of a matrix A, the condition numberof λj is defined as

κ(λj) =∥uj∥∥vj∥∣u∗j vj ∣

,

where vj and u∗j are the right and left eigenvectors associated with λj , respectively.

Note: The Cauchy-Schwarz inequality implies that ∣u∗j vj ∣ ≤ ∥uj∥∥vj∥, so κ(λj) ≥ 1,

with equality when uj and vj are collinear. An eigenvalue for which κ(λj) = 1 iscalled a normal eigenvalue; a matrix A with all eigenvalues simple is normal if andonly if κ(λj) = 1 for all eigenvalues.

With this definition, we can find finer bounds for the pseudospectrum of a matrix; inparticular, we can find bounds for the components of the pseduospectrum centeredaround each eigenvalue.

The following theorem can be found for example in [3].

Theorem 2.5 (Asymptotic pseudospectra inclusion regions). Suppose A ∈ CN×N

has N distinct eigenvalues. Then, as ε→ 0,

σε(A) ⊆N

⋃j=1

B (λj , εκ(λj) +O (ε2)) .

We can drop the O(ε2) term, for which we get an increase in the radius of ourinclusion disks by a factor of N [2, Thm. 4].

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10 GONG, MEYERSON, MEZA, WARD, STOICIU

Theorem 2.6 (Bauer-Fike theorem based on κ(λj)). Suppose A ∈ CN×N has Ndistinct eigenvalues. Then ∀ε > 0,

σε(A) ⊆N

⋃j=1

B (λj , εNκ(λj)) .

The following has a proof in [14].

Theorem 2.7 (Asymptotic formula for the resolvent norm). Let λj ∈ σ(A) be aneigenvalue of index kj. For any z ∈ σε(A), for small enough ε,

∣z − λj ∣ ≤ (Cjε)1kj ,

where Cj = ∥VjTkj−1j U∗

j ∥ and T = J − λI.

3. Pseudospectra of 2 × 2 Matrices

The following section presents a complete characterization of the ε-pseudospectrumof any 2 × 2 matrix. We classify 2 × 2 matrices by whether they are diagonalizableor non-diagonalizable and then determine the ε-pseudospectra for each class. Webegin with an explicit formula for computing the norm of a 2 × 2 matrix, and thenanalyze a 2 × 2 Jordan block.

Let A = (a bc d

), with a, b, c, d ∈ C. Let smax denote the largest singular value of A.

Then,

∣∣A∣∣2 = smax

= largest eigenvalue of A∗A

=Tr(A∗A) +

√Tr(A∗A)2 − 4 det(A∗A)

2.

Letting ρ = ∣a∣2 + ∣b∣2 + ∣c∣2 + ∣d∣2, we can expand the norm as

∥A∥2 = 1

2(√ρ2 − 4 (∣a∣2∣d∣2 + ∣b∣2∣c∣2 − (adbc + adbc)) + ρ) .

This formula for the norm of the matrix allows us to compute the ε-pseudospectraof 2 × 2 matrices.

3.1. Non-diagonalizable 2 × 2 Matrices.

Proposition 3.1. Let A be any non-diagonalizable 2 × 2 matrix, and let λ denotethe eigenvalue of A. Write A = V JV −1 where

V = (a bc d

) , a, b, c, d ∈ C, and J = (λ 10 λ

) .

Then σε(A) is exactly a disk. More specifically, given any ε,

σε(A) = B (λ, ∣k∣)

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EXPLICIT BOUNDS FOR THE PSEUDOSPECTRA OF MATRICES AND OPERATORS 11

where

(3.1) ∣k∣ =√Cε + ε2 and C = ∣a∣2 + ∣c∣2

∣ad − bc∣.

Proof. Let z = λ + k where k ∈ C. Then,

(z −A)−1 = (z − V JV −1)−1 = V (z − J)−1V −1.

Let k = z − λ. We obtain that

∥(z −A)−1∥ = 1

∣k2(ad − bc)∣∥(adk − ac − bck a2

−c2 −bck + ac + adk)∥ .

Now, let

M = (adk − ac − bck a2

−c2 −bck + ac + adk) .

A simple calculation gives

Tr(M∗M) = (∣a∣2 + ∣c∣2)2 + 2∣k∣2∣bc − ad∣2

det(M∗M) = ∣k∣4∣bc − ad∣4

From the formula for the norm, we obtain that

∥(z −A)−1∥ = ∥V (z − J)−1V −1∥

=

√Tr(M∗M) +

√Tr(M∗M)2 − 4 det(M∗M)

∣k∣2∣ad − bc∣√

2.

Note that this function depends only on ∣k∣ = ∣z − λ∣; thus for any ε, σε(A) willbe a disk. Solving for k in the above equation to find the curve bounding thepseudospectrum, we obtain

∣k∣ =

¿ÁÁÀε

∣a∣2 + ∣c∣2∣ad − bc∣

+ ε2.

Example 3.1. Perhaps the simplest example of such a matrix is the 2 × 2 Jordanblock.

Let J = (λ 10 λ

) where λ ∈ C. In Jordan form, J decomposes as

J = (1 00 1

)(λ 10 λ

)(1 00 1

) .

Plugging into Equation 3.1 with a = 1, b = 0, c = 0, d = 1, we get that

∣k∣ =√ε2(1 − 0)(1 − 0) + ε∣1 − 0∣(1 + 0)

∣1 − 0∣=√ε + ε2.

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12 GONG, MEYERSON, MEZA, WARD, STOICIU

Hence,

σε(J) = B (λ,√ε + ε2) .

We can check this result by direct application of the first definition of pseudospectra.

Let z ∈ σε(J). Note

(z − J)−1 = (z − λ −10 z − λ)

−1

= 1

(z − λ)2(z − λ 1

0 z − λ) .

Using our equation for the norm of a 2 × 2 matrix, we get

1

ε< ∣∣(z − J)−1∣∣ = 1

∣z − λ∣2

√1

2(2∣z − λ∣2 + 1 +

√4∣z − λ∣2 + 1).

Let w = ∣z−λ∣, and square both sides to get 2w4

ε2< 2w2+1+

√4w2 + 1. This simplifies

to w <√ε + ε2. Thus, z ∈ σε(J) ⇐⇒ ∣z − λ∣ <

√ε + ε2 ⇐⇒ z ∈ B (λ,

√ε + ε2).

3.2. Diagonalizable 2 × 2 Matrices.

Proposition 3.2. Let A be any diagonalizable 2 × 2 matrix and let λ1, λ2 be theeigenvalues of A and v1, v2 be the eigenvectors associated with the eigenvalues. Thenσε(A) is the set of points z that satisfy the equation

(ε2 − ∣z − λ1∣2)(ε2 − ∣z − λ2∣2) − ε2∣λ1 − λ2∣2 cot2(θ) < 0,

where θ is the angle between the two eigenvectors.

Proof. Since A is diagonalizable, we know that A = V DV −1 where

V = (a bc d

) D = (λ1 00 λ2

) .

Note that V is an invertible matrix with columns corresponding to the eigenvectorsof A.

Without loss of generality, we can write z = λ1 + k.

Let γ = λ1 − λ2 and r = ad − bc. Then,

(z −A)−1 = (z − V DV −1)−1 = V (z −D)−1V −1

= 1

r(a bc d

)(1k

00 1

γ+k

)( d −b−c a

)

= ( 1

rk(γ + k))(adγ + rk −abγ

cdγ −bcγ + rk) .

Let M = (adγ + rk −abγcdγ −bcγ + rk).

Then, ∥(z −A)−1∥ = 1∣rk(γ+k)∣

∥M∥.

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EXPLICIT BOUNDS FOR THE PSEUDOSPECTRA OF MATRICES AND OPERATORS 13

Calculating Tr(M∗M) and det(M∗M) gives

Tr(M∗M) = ∣cdγ∣2 + ∣kr + adγ∣2 + ∣abγ∣2 + ∣kr − bcγ∣2

= 2∣kr∣2 + 2∣r∣2 Re[kγ] + ∣γ∣2(∣a∣2 + ∣c∣2)(∣b∣2 + ∣d∣2)

= ∣r∣2 (2∣k∣2 + 2 Re[kγ] + ∣γ∣2 + ∣γ∣2 ((∣a∣2 + ∣c∣2)(∣b∣2 + ∣d∣2)∣r∣2

− 1)) .

Note that the first part of this expression simplifies to

2∣k∣2 + 2 Re[kγ] + ∣γ∣2 = ∣k + γ∣2 + ∣k∣2.For the second part, we have

(∣a∣2 + ∣c∣2)(∣b∣2 + ∣d∣2)∣r∣2

− 1 = 1

∣r∣2((∣a∣2 + ∣c∣2)(∣b∣2 + ∣d∣2) − ∣r∣2)

= 1

∣r∣2(∣ab∣2 + ∣cd∣2 + bcad + bcad).

Now, note that our eigenvectors v1, v2 are exactly the columns of V . So, we have

∣v1 ⋅ v2∣2 = ∣ab + cd∣2 = ∣ab∣2 + ∣cd∣2 + adbc + adbc.Thus, the second term in our trace simplifies to

∣v1 ⋅ v2∣2

∣r∣2= ∣v1∣2∣v2∣2 cos2 θ

∣v1∣2∣v2∣2 sin2 θ= cot2 θ,

where the denominator comes from the fact that r = detV is the area of the paral-lelogram spanned by v1 and v2. Thus, we get that

Tr(M∗M) = ∣r∣2 (∣γ + k∣2 + ∣k∣2 + ∣γ∣2 cot2 θ) .For the determinant, we have

det(M∗M) = (∣cdγ∣2 + ∣kr + adγ∣2)(∣abγ∣2 + ∣kr − bcγ∣2)

− ((kr − bcγ)cdγ − abγ(kr + adγ))(−(kr + adγ)abγ + cdγ(kr − bcγ))

= (k2r2 + kr2γ)(k2r2 + kr2γ)= ∣r∣4∣k∣2∣k + γ∣2

We then plug these in to the equation for the norm of a 2 × 2 matrix and get

ε−1 < ∥(z −A)−1∥ =

¿ÁÁÀTr(M∗M) +

√Tr(M∗M)2 − 4 det(M∗M)

2∣r∣2∣k(γ + k)∣2

⇐⇒ 2∣k(γ + k)∣2

ε2< ∣γ + k∣2 + ∣k∣2 + ∣γ∣2 cot2 θ +

√(∣γ + k∣2 + ∣k∣2 + ∣γ∣2 cot2 θ)2 − 4∣k(γ + k)∣2

⇐⇒ (2∣k(γ + k)∣2

ε2− (∣γ + k∣2 + ∣k∣2 + ∣γ∣2 cot2 θ))

2

< (∣γ + k∣2 + ∣k∣2 + ∣γ∣2 cot2 θ)2 − 4∣k(γ + k)∣2

⇐⇒ 4∣k(γ + k)∣4

ε4− 4∣k(γ + k)∣2

ε2(∣γ + k∣2 + ∣k∣2 + ∣γ∣2 cot2 θ) + 4∣k(γ + k)∣2 < 0

⇐⇒ 4∣k(γ + k)∣2 + ε4 − (∣γ + k∣2 + ∣k∣2 + ∣γ∣2 cot2 θ)ε2 < 0

⇐⇒ (ε2 − ∣k∣2)(ε2 − ∣k + γ∣2) − ε2∣γ∣2 cot2 θ < 0.

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14 GONG, MEYERSON, MEZA, WARD, STOICIU

Note that for normal matrices, the eigenvectors are orthogonal, so θ = π/2 andcot θ = 0. Therefore the equation above reduces to

(ε2 − ∣k∣2)(ε2 − ∣k + γ∣2) < 0

which describes two disks of radius ε centered around λ1, λ2, as we expect.

When the matrix only has one eigenvalue and is still diagonalizable (i.e. when it isa multiple of the identity), then we get

(ε2 − ∣k∣2)(ε2 − ∣k + 0∣2) < 0,

which is a disk of radius ε centered around the eigenvalue.

One consequence to note of Proposition 3.2 is that the shape of σε(A) is dependenton both the eigenvalues and the eigenvectors of A. Another less obvious consequenceis that the pseudospectrum of a 2×2 matrix approaches a union of disks as ε tendsto 0. This is proven in the following proposition.

Proposition 3.3. Let A be a diagonalizable 2 × 2 matrix. Then, σε(A) asymptot-ically tends toward a disk. In particular, rmax

rmin= 1 +O(ε), where rmax, rmin are the

maximum and minimum distances from an eigenvalue to the boundary of the closerconnected component of ∂σε(A). Moreover, for A diagonalizable but not normal,σε(A) is never a perfect disk.

Proof. Let ε be small enough so that the ε-pseudospectrum is disconnected. With-out loss of generality, we will show that the component around λ1 tends towardsa disk by showing that the ratio of the maximum distance from z to λ1 to theminimum distance from z to λ1 is 1 + o(ε).

Let zmax ∈ ∂σε(A) such that ∣zmax −λ1∣ is a maximum. Consider the line joining λ1and λ2. Suppose for contradiction that zmax did not lie on this line. Then, rotatezmax in the direction of λ2 so that it is on this line, and call this new point z′. Notethat ∣z′ − λ2∣ < ∣zmax − λ2∣, but ∣z′ − λ1∣ = ∣zmax − λ1∣. As such, we get that

(∣z′−λ1∣2−ε2)(∣z′−λ2∣2−ε2) < (∣zmax−λ1∣2−ε2)(∣zmax−λ2∣2−ε2) = ε2∣λ1−λ2∣2 cot2 θ

Thus, from Proposition 3.2 z′ ∈ σε(A) but z′ is not on the boundary of σε(A).Starting from z′ and traversing the line joining λ1 and λ2, we can find z′′ ∈ ∂σε(A)such that ∣z′′ − λ1∣ > ∣z′ − λ1∣ = ∣zmax − λ1∣. This contradicts our choice of zmax andso zmax must be on the line joining λ1 and λ2. A similar argument shows that zmin

must also be on the line joining λ1 and λ2, where zmin ∈ ∂σε(A) such that ∣zmin−λ1∣is a minimum.

Since zmax is on the line joining λ1 and λ2, we have the exact equality ∣zmax −λ2∣ =∣zmax − λ1∣ + ∣λ2 − λ1∣. Let rmax = ∣zmax − λ1∣ and let y = ∣λ2 − λ1∣. The equationdescribing rmax becomes

(r2max − ε2) ((y − rmax)2 − ε2) = ε2y2 cot2 θ

Solving this equation for rmax, we get

rmax =1

2(y −

√y2 + 4ε2 − 4yε csc θ)

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EXPLICIT BOUNDS FOR THE PSEUDOSPECTRA OF MATRICES AND OPERATORS 15

Similarly, we get

rmin =1

2(√y2 + 4ε2 + 4yε csc θ − y)

For ε small, we can use the approximation (1+ε)p = 1+pε+ p(p−1)2

ε2 +o(ε3). Usingthis, we see

rmax

rmin=

1 −√

1 + 4(ε/y)2 − 4(ε/y) csc θ√

1 + 4(ε/y)2 + 4(ε/y) csc θ − 1

= 2(ε/y) csc θ − 2(ε/y)2 + 2(ε/y)2 csc2 θ +O(ε3)2(ε/y) csc θ + 2(ε/y)2 − 2(ε/y)2 csc2 θ +O(ε3)

= csc θ + (ε/y)(csc2 θ − 1) +O(ε2)csc θ − (ε/y)(csc2 θ − 1) +O(ε2)

= 1 + ηε +O(ε2)1 − ηε +O(ε2)

where η = cos θ cot θ. Recall the geometric series approximation 11−x

= 1+x+O(x2).Using this, we get our desired result

rmax

rmin= (1 + ηε +O(ε2))(1 + ηε +O(ε2))

= 1 + 2ηε +O(ε2)= 1 + (2 cos θ cot θ)ε +O(ε2)

Thus, as ε→ 0, the farthest point on the boundary of σε(A) from λ1 tends towardsthe closest point on the boundary of σε(A) from λ1. Hence, σε(A) tends towardsa disk.

Moreover, if A is diagonalizable but not normal, then the eigenvectors are linearlyindependent but not orthogonal, so θ is not a multiple of π/2 or π, and hencecos θ cot θ ≠ 0 and rmax

rmin≠ 1.

4. Asymptotic Union of Disks Theorem

In the previous section, we showed that the ε-pseudospectrum for all 2 × 2 ma-trices are disks or asymptotically converge to a union of disks. We now explorewhether this behavior holds in the general case. It is possible to find matriceswhose ε−pseudospectra exhibit pathological properties for large ε; for example, thefollowing non-diagonalizable matrix has, for larger ε, an ε-pseudospectrum that isnot convex and not simply connected.

Thus, pseudospectra may behave poorly for large enough ε; however, in the limit asε → 0, these properties disappear and the pseudospectra behave as disks centeredaround the eigenvalues with well-understood radii. The following theorem allowsus to understand this asymptotic behavior.

We will use the following set-up (which follows [10]) for our theorem.

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16 GONG, MEYERSON, MEZA, WARD, STOICIU

⎛⎜⎜⎜⎜⎜⎝

−1 −10 −100 −1000 −100000 −1 −10 −100 −10000 0 −1 −10 −1000 0 0 −1 −100 0 0 0 −1

⎞⎟⎟⎟⎟⎟⎠

Figure 4.1. Pseudospectra of a Toeplitz matrix

For A ∈ CN×N , find matrices that Jordanize A. That is,

⎛⎝J

J

⎞⎠=⎛⎝Q

Q

⎞⎠A (P P ) ,

⎛⎝Q

Q

⎞⎠(P P ) = I

and J consists of Jordan blocks J1, . . . , Jm corresponding to an eigenvalue λ. J isthe part of the Jordan form containing the other eigenvalues of A.

Let n be the size of the largest Jordan block corresponding to λ, and suppose thereare ` Jordan blocks of size n × n. Arrange J1, . . . , Jm in J so that

dim (J1) = ⋯ = dim(J`) > dim (J`+1) ≥ ⋯ ≥ dim (Jm)

where J1, . . . , J` are n × n.

Further partition

P = (P1 , . . . , P` , . . . , Pm)in a way that agrees with the above partition of J , so that the first column, xj , ofeach Pj is a right eigenvector of A associated with λ. We also partition Q likewise

Q =

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝

Q1

⋮Q`

⋮Qm

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠

.

The last row, yj , of each Qj is a left eigenvector of A corresponding to λ.We now build the matrices

Y =

⎛⎜⎜⎜⎜⎜⎜⎝

y1

y2

⋮y`

⎞⎟⎟⎟⎟⎟⎟⎠

, X = (x1 , x2 , . . . , x`) ,

where X and Y are the matrices of right and left eigenvectors, respectively, corre-sponding to the Jordan blocks of maximal size for λ.

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EXPLICIT BOUNDS FOR THE PSEUDOSPECTRA OF MATRICES AND OPERATORS 17

With these matrices defined, we can understand the radii of the ε-pseudospectrumof A as ε→ 0.

Theorem 4.1. Let A ∈ CN×N . Let ε > 0. Then given λ ∈ σ(A), for ε small enough,there exists a connected component U ⊆ σε(A) such that U ∩ σ(A) = λ; denote thiscomponent of the ε-pseudospectrum σε(A) ↾λ.

Then, as ε→ 0,

B(λ, (Cε)1/n + o(ε1/n)) ⊆ σε(A) ↾λ⊆ B(λ, (Cε)1/n + o(ε1/n))where C = ∥XY ∥, with X,Y defined above.

Proof.

Lower Bound: We begin by proving the first containment,

B(λ, (Cε)1/n + o(ε1/n)) ⊆ σε(A) ↾λ .Let E ∈ CN×N . From [10, Theorem 2.1], we know there is an eigenvalue of theperturbed matrix A + εE admitting a first order expansion

λ(ε) = λ + (γε)1/n + o(ε1/n),where γ is any eigenvalue of Y EX. [10] later shows in Theorem 4.2 that

α ∶= max∥E∥≤1

γmax = ∥XY ∥,

where γmax is the largest eigenvalue of Y EX. In fact, the E that maximizes γ isgiven by E = vu where v and u are the right and left singular vectors of the largestsingular value of XY , normalized so ∥v∥ = ∥u∥ = 1.

We claim that B(λ, (∣α∣ε)1/n + o(ε1/n)) ⊆ σε(A).

Fix θ ∈ [0,2π], and define E = eiθE. Considering the perturbed matrix A + εE, wecan write

λ(ε) = λ + (γε)1/n + o(ε1/n)where γ is now any eigenvalue of Y EX. Solving for γ, we obtain:

0 = det(eiθY EX − γI) = det (eiθ(Y EX − e−iθγI)) ,

which implies 0 = det(Y EX − e−iθγI), and thus e−iθγ is any eigenvalue of Y EX.Since α is an eigenvalue of Y EX, we can take γ = eiθα. Thus, we obtain

λ + (eiθαε)1/n + o(ε1/n) ∈ σε(A).Ranging θ from 0 to 2nπ, we get

B(λ, (∣α∣ε)1/n + o(ε1/n)) ⊆ σε(A).

Upper Bound: We now prove the second containment,

σε(A) ↾λ⊆ B(λ, (Cε)1/n + o(ε1/n)).

From [14, §52, Theorem 52.3], we have that asymptotically

σε(A) ↾λ⊆ B(λ, (βε)1/n + o(ε1/n)),where β = ∥PDn−1Q∥ and J = λI +D. We claim β = ∥XY ∥ = α.

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18 GONG, MEYERSON, MEZA, WARD, STOICIU

Note that Dn−1 = diag[Γ1, . . .Γ`,0] where Γk is a n × n matrix with a 1 in the topright entry and zeros elsewhere. We find

PDn−1 =

⎛⎜⎜⎜⎜⎜⎝

x1 x2 ⋯ x`

⎞⎟⎟⎟⎟⎟⎠

.

This then gives

PDn−1Q = (XY 00 0

) .

Thus β = ∥PDn−1Q∥ = ∥XY ∥ = α.

Combining this with our lower bound result, we see that as ε→ 0,

σε(A) ↾λ= B(λ, (αε)1/n + o(ε1/n)).

We present special cases of matrices to explore the consequences of Theorem 4.1.

Special Cases:

(1) λ is simple. That is, λ has algebraic multiplicity 1.Then, n = 1 and X and Y become the right and left eigenvectors x and

y∗ for λ, respectively. Hence, C = ∥XY ∥ = ∥xy∗∥ = ∥x∥∥y∥ = κ(λ) where wenormalize so that ∣y∗x∣ = 1. Then, Theorem 4.1 becomes

σε(A) ↾λ≈ B(λ,κ(λ)ε)

which matches with Theorem 2.5.(2) λ is semisimple. That is, the algebraic multiplicity of λ equals the geometric

multiplicity.Then, n = 1, and we obtain X = P and Y = Q, and C becomes the norm

of the spectral projector for λ; see [14].(3) λ has geometric multiplicity 1

In this case we obtain the same result for when λ is simple, except nmay not equal 1. In other words,

σε(A) ↾λ≈ B(λ, (κ(λ)ε)1/n).

(4) A ∈ C2×2.There are two cases, as in Section 3: Pseudospectra of 2 × 2 Matrices.

First, assume A is non-diagonalizable. In this case, A only has one eigen-value, λ. Let

V = (a bc d

) , V −1 = 1

ad − bc( d −b−c a

)

where A = V JV −1.

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EXPLICIT BOUNDS FOR THE PSEUDOSPECTRA OF MATRICES AND OPERATORS 19

Then, we have that

X = (ac) , Y = 1

ad − bc(−c a) .

From Theorem 4.1, we then have that as ε→ 0,

σε(A) ⊆ B⎛⎝λ,( ∣a∣2 + ∣c∣2

∣ad − bc∣ε)

1/2

+ o (ε1/2)⎞⎠.

Equation 3.1 gives the explicit formula for σε(A) as ε→ 0, namely

σε(A) = B⎛⎝λ,( ∣a∣2 + ∣c∣2

∣ad − bc∣ε + ε2)

1/2⎞⎠

= B⎛⎝λ,( ∣a∣2 + ∣c∣2

∣ad − bc∣ε)

1/2

+O(ε3/2)⎞⎠,

which agrees with the above theorem.

In the case where A is diagonalizable, A has two eigenvalues, λ1 and λ2.Without loss of generality, we will look at σε(A) around λ1.

A = (a bc d

)(λ1 00 λ2

)[ 1

ad − bc( d −b−c a

)] .

From this, we have

X = (ac) , Y = 1

ad − bc(d −b) .

∥XY ∥ =(∣a∣2 + ∣c∣2) (∣b∣2 + ∣d∣2)

∣ad − bc∣= csc θ.

Thus, as ε→ 0, we have from Theorem 4.1:

B (λ, (csc θ) ε + o (ε)) ⊆ σε(A) ⊆ B (λ, (csc θ) ε + o (ε)) .

So,

rmax

rmin= (csc θ) ε + o (ε)

(csc θ) ε + o (ε).

Note as ε→ 0, that this tends to 1 faster than ε.This agrees with the ratio we obtain from the explicit formula for diag-

onalizable 2 × 2 matrices. However, the formula for the 2 × 2 matrix givesus slightly more information on the o(ε) term.

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20 GONG, MEYERSON, MEZA, WARD, STOICIU

rmax =1

2(y −

√y2 + 4ε2 − 4yε csc θ)

rmin =1

2(√y + 4ε2 + 4yε csc θ − y)

rmax

rmin=

1 −√

1 + 4(ε/y)2 − 4(ε/y) csc θ√

1 + 4(ε/y)2 + 4(ε/y) csc θ − 1

= 2(ε/y) csc θ − 2(ε/y)2 + 2(ε/y)2 csc2 θ + o(ε3)2(ε/y) csc θ + 2(ε/y)2 − 2(ε/y)2 csc2 θ + o(ε3)

= csc θ + (ε/y)[csc2 θ − 1] + o(ε2)csc θ − (ε/y)[csc2 θ − 1] + o(ε2)

.

5. Pseudospectra of N ×N Jordan Block

As the Jordan blocks are the fundamental structural elements of finite dimensionallinear operators, it is especially important to understand their pseudospectra.

Proposition 5.1 (Lower Bound). Let J be an N ×N Jordan block where

J =

⎛⎜⎜⎜⎜⎜⎝

λ 1λ 1

⋱ ⋱⋱ 1

λ

⎞⎟⎟⎟⎟⎟⎠

.

Then,

B(λ, N√ε) ⊆ σε(J).

Proof. By induction. This is clear for N = 1; for N = 2, we will show that

B(λ,√ε) ⊆ σε(J).

Let z ∈ B(λ,√ε). Write z = λ + ω, where ∣ω∣ <

√ε. Define

E = ( 0 0ω2 0

) .

Note ∣∣E∣∣ = ∣ω∣2 < ε. Observe

det(J +E − zI) = det(λ − z 1ω2 λ − z) = det(−ω 1

ω2 −ω) = 0.

Thus, z ∈ σ(J +E), and hence z ∈ σε(J).

Now, assume that the proposition is true for all N ×N Jordan matrices. Let J bean (N + 1) × (N + 1) Jordan matrix, let z ∈ B (λ, εN+1), and let z = λ + ω, where

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EXPLICIT BOUNDS FOR THE PSEUDOSPECTRA OF MATRICES AND OPERATORS 21

∣ω∣ < εN+1. Let E be a matrix with ωN+1 in the bottom-left corner entry and zeroselsewhere. Note that ∥E∥ = ∣ω∣N+1 < ε. Then,

det(J +E − zI) = (−ω)N+1 + (−1)NωN+1 = ωN+1 ((−1)N+1 + (−1)N) = 0.

Thus, z ∈ σ(J +E), and so z ∈ σε(J). �

From the second definition of the pseudospectrum, using a perturbation matrix, wecan get a better explicit lower bound on the ε-pseudospectra of an N ×N Jordanblock.

Proposition 5.2 (Better Lower Bound). Let J be an N ×N Jordan block. Then,

B (λ, N√ε(1 + ε)N−1) ⊆ σε(J).

Proof. We use the second definition for σε(J). Let

E =

⎛⎜⎜⎜⎜⎜⎝

0 k0 k

⋱ ⋱⋱ k

k 0

⎞⎟⎟⎟⎟⎟⎠

where ∣k∣ < ε, and note that ∥E∥ < ε. We take det(J +E − zI) and set it equal tozero to find the eigenvalues of J +E.

0 = det(J +E − zI)

= det

⎛⎜⎜⎜⎜⎜⎝

λ − z k + 1λ − z k + 1

⋱ ⋱⋱ k + 1

k λ − z

⎞⎟⎟⎟⎟⎟⎠

= (λ − z)N + (−1)N−1k(1 + k)N−1

= (−1)N−1((z − λ)N + k(1 + k)N−1);

⇐⇒ (z − λ)N = k(1 + k)N−1

z − λ = N√k(1 + k)N−1.

So, B (λ, N√ε(1 + ε)N−1) ⊆ σε(J). �

From [14, pg. 470], we know that the ε-pseudospectrum of the Jordan block is aperfect disk about the eigenvalue of J of some radius. Although we do not find theexplicit radius, we can use Theorem 4.1 to find the asymptotic behavior of σε(J),

Proposition 5.3 (Asymptotic Bound). Let J be an N ×N Jordan block. Then

σε(J) ≈ B(λ, ε1/N + o(ε1/N)).

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22 GONG, MEYERSON, MEZA, WARD, STOICIU

Proof. The N × N Jordan block has left and right eigenvectors uj and vj where∥uj∥ = 1 and ∥vj∥ = 1. So, from Theorem 4.1, we find C = ∥XY ∥ = ∥vjuj∥ = 1. Thus,

σε(J) ≈ B(λ, ε1/N + o(ε1/N)).

However, we can find the perturbation matrix that gives us the boundary of σε(J).

Note: Let J be an n × n Jordan block and E be the perturbation matrix

E =

⎛⎜⎜⎜⎜⎜⎝

ε⋰

⋰⋰

ε

⎞⎟⎟⎟⎟⎟⎠

.

We know that the ε-pseudospectrum of any Jordan block is a perfect disk aroundthe eigenvalue. We observe that as we range θ from 0 to 2π, σ(A+eiθE) numericallytraces the boundary of σε(J) for ε > 0.

For example, suppose J is a 5 × 5 Jordan block with 0 as an eigenvalue. Lettingε = .025, we get through eigtool that the pseudospectral radius (defined as thefarthest point on the boundary of pseudo spectrum from 0) is 0.50765. We now useMathematica to compute the eigenvalue of (J +E) and obtain 0.50765 as well.

6. Pseudospectra of bidiagonal matrices

The Jordan block is a simple example of a bidiagonal matrix. We now extend ourapplication of Theorem 4.1 to all bidiagonal matrices.

6.1. Asymptotic Bounds for Periodic Bidiagonal Matrices.

We investigate various classes of bidiagonal matrices. We describe the asymptoticbehavior of the ε-pseudospectrum for the general periodic bidiagonal case, namelyA is an N×N matrix with period k on the main diagonal and nonzero superdiagonalentries

A =

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

a1 b1⋱ ⋱

ak ⋱⋱ ⋱

⋱ ⋱a1 ⋱

⋱ bN−1

ar

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

We have from Theorem 4.1, that

σε(A) ≈k

⋃j=1

B (aj , (Cjε)1nj ) ,

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EXPLICIT BOUNDS FOR THE PSEUDOSPECTRA OF MATRICES AND OPERATORS 23

where nj is the size of the Jordan block corresponding to aj and Cj = ∥XjYj∥.For matrices that are of the bidiagonal form, the geometric multiplicity of eacheigenvalue is 1, meaning that all the eigenvalues fall in Case 3 of Theorem 4.1.Thus, the constant Cj that multiplies any eigenvalue aj is simply Cj = ∥vj∥∥uj∥,where vj and uj denote the right and left eigenvectors, respectively.

We will begin by introducing ε-pseudospectrum for simple special cases which leadto the most general case.

The cases will be presented as follows:

(1) Let A be a kn × kn matrix with a1, . . . , ak distinct and bi = 1 for all i.(2) Let A be a kn × kn matrix with a1, . . . , ak distinct.(3) Let A be an N ×N matrix with a1, . . . , ak distinct.(4) General Case: Let A be an N ×N matrix with a1, . . . , ak not distinct.

We define:

f(x) =⎧⎪⎪⎨⎪⎪⎩

x, x ≠ 0

1, x = 1.

Case 1:

● The size of A is kn × kn.● The ai’s are distinct.● bi = 1 for all i.

So, A is of the form:

A =

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

a1 1⋱ ⋱

ak ⋱⋱ ⋱

⋱ ⋱a1 ⋱

⋱ 1ak

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

.

To find the asymptotic bound for the resolvent norm of A around each eigenvalue,we must determine the right and left eigenvectors associated with each eigenvalue.We claim:

vj =

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

1f(aj−aj−1)⋯f(aj−a1)

1f(aj−aj−1)⋯f(aj−a2)

⋮1

f(aj−aj−1)

10⋮0

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

,

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24 GONG, MEYERSON, MEZA, WARD, STOICIU

uj =f(a1 − aj)⋯f(aj−1 − aj)

[f(a1 − aj)⋯f(aj−1 − aj)f(aj+1 − aj)⋯f(ak − aj)]n

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

0⋮0

f(aj+1 − aj)⋯f(ak − aj)f(aj+2 − aj)⋯f(ak − aj)

⋮f(ak − aj)

1

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

,

which can be verified by direction computation of the left and right eigenvectorsfor each eigenvalue a1, . . . , ak.

We will check that vj is a right eigenvector for eigenvalue aj by cases.

First observe that for i > j, we have (vj)i = 0. Therefore,

(Avj)i = Aii(vj)i +Ai(i+1)(vj)i+1 = 0 = aj(vj)i.

Now suppose i ≤ j. Then, we have

(Avj)i = Aii(vj)i +Ai(i+1)(vj)i+1 = ai(vj)i + (vj)i+1.

For 1 ≤ i ≤ j − 2, we get

(Avj)i = ai ⋅1

(aj − aj−1)⋯(aj − ai)+ 1

(aj − aj−1)⋯(aj − ai+1)

=ai + (aj − ai)

(aj − aj−1)⋯(aj − ai)= aj(vj)i.

For i = j − 1, we get

(Avj)j−1 = aj−1 ⋅1

aj − aj−1+ 1 = aj(vj)j−1.

For i = j we get

(Avj)j = aj(vj)j + 0.

Thus, in all cases we see that Avj = ajvj , and so vj is indeed a right eigenvectorfor the eigenvalue aj . The proof that u∗j is a left eigenvector for eigenvalue aj issimilar.

Case 2:

● The size of A is kn × kn.● The ai’s are distinct.

Since the superdiagonal of A is not constant, we now write the elements of thesuperdiagonal as b1, b2, . . . , bN−1.

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EXPLICIT BOUNDS FOR THE PSEUDOSPECTRA OF MATRICES AND OPERATORS 25

We claim:

vj =

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

b1⋯bj−1f(aj−a1)⋯f(aj−aj−1)

b2⋯bj−1f(aj−a2)⋯f(aj−aj−1)

⋮bj−1

f(aj−aj−1)

10⋮0

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

,

u∗j = (0, ⋯ 0, y,bp

f(aj+1−aj)y, ⋯ bp⋯bN−2

f(aj+1−aj)⋯f(ak−1−aj)y,

bp⋯bN−1f(aj+1−aj)⋯f(ak−aj)

y) ,

where p = k(n − 1) + j, and

y =bj⋯bp−1

((a1 − aj)⋯(aj−1 − aj)(aj+1 − aj)⋯(ak − aj))n−1.

Again, direct computation will show that these are indeed left and right eigenvectorsassociated with any eigenvalue aj .

Case 3:

● The size of A is N ×N .● The ai’s are distinct.

We now drop our assumption that the size of our matrix is kn×kn, for period k onthe diagonal. Let n, r be such that N = kn + r, where 0 < r ≤ k. In other words, aris the last entry on the main diagonal, so the period does not necessarily complete.

For aj , the right eigenvector is given by

vj =

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

b1⋯bj−1f(aj−a1)⋯f(aj−aj−1)

b2⋯bj−1f(aj−a2)⋯f(aj−aj−1)

⋮bj−1

f(aj−aj−1)

10⋮0

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

.

We split up the formula for the left eigenvectors into two cases:

(1) 1 ≤ j ≤ r(2) r < j ≤ k.

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26 GONG, MEYERSON, MEZA, WARD, STOICIU

(1) 1 ≤ j ≤ rOn the main diagonal, there are n complete blocks with entries a1, . . . , ak, and onepartial block at the end with entries a1, . . . ar. In the first case, when 1 ≤ j ≤ r,then aj is in this last partial block. In this case then, let p = kn + j.

We claim

uj = µj

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

0⋮0

f(aj+1 − aj)⋯f(ar − aj)bpf(aj+2 − aj)⋯f(ar − aj)

⋮bp⋯bN−2f(ar − aj)

bp⋯bN−1

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

,

where µj = bj⋯bp−1f(a1−aj)⋯f(aj−1−aj)

[f(a1−aj)⋯f(aj−1−aj)f(aj+1−aj)⋯f(ak−aj)]nf(a1−aj)⋯f(ar−aj).

(2) r < j ≤ kIn this case, aj is in the last complete block. Now, let p = k(n − 1) + j.

Now, we claim

uj = µj

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

0⋮0

f(a1 − aj)⋯f(ar − aj)f(aj+1 − aj)⋯f(ak − aj)bpf(a1 − aj)⋯f(ar − aj)f(aj+2 − aj)⋯f(ak − aj)

⋮bp⋯bp+k−j−1f(a1 − aj)⋯f(ar − aj)bp⋯bp+k−jf(a2 − aj)⋯f(ar − aj)

⋮bp⋯bN−2f(ar − aj)

bp⋯bN−1

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

,

again where µj = bj⋯bp−1f(a1−aj)⋯f(aj−1−aj)

[f(a1−aj)⋯f(aj−1−aj)f(aj+1−aj)⋯f(ak−aj)]nf(a1−aj)⋯f(ar−aj)

Case 4: General Case.

● The size of A is N ×N .● The ai’s are not distinct for 1 ≤ i ≤ k

Let A be a N ×N periodic bidiagonal matrix with period k on the main diagonal.Let n, r be such that N = kn + r, where 0 < r ≤ k. Write a1, . . . , ak for the entrieson the main diagonal (ai’s not distinct) and b1, . . . , bN−1 for the entries on thesuperdiagonal. Let ar be the last entry on the main diagonal.

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EXPLICIT BOUNDS FOR THE PSEUDOSPECTRA OF MATRICES AND OPERATORS 27

We can explicitly find the left and right eigenvectors for any eigenvalue, α. Sup-pose α first appears in position ` of the period k. Then the corresponding righteigenvector for α is the same form as v` in Case 4. That is,

v` =

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

b1⋯b`−1f(a`−a1)⋯f(a`−a`−1)

b2⋯b`−1f(a`−a2)⋯f(a`−a`−1)

⋮b`−1

f(a`−a`−1)

10⋮0

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

.

The corresponding left eigenvector for α depends on the first and last positions ofα. Let q ≡ m (mod k). We split up the formula for the left eigenvector of α intotwo cases, which again mirror the formulas given in Case 4:

(1) 1 ≤ ` ≤ r(2) r < ` ≤ k.

For both of these two cases, we define

f(bi) =⎧⎪⎪⎨⎪⎪⎩

bi, i ≥ p1, i < p

.

(1) 1 ≤ ` ≤ rIn this case then, α appears in the partial block. Let p = kn + `.

We claim

u` = µ`

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

0⋮0

f(bp+m−`−1)f(am+1 − a`)⋯f(ar − a`)f(bp+m−`−1)f(bp+m−l)f(am+2 − a`)⋯f(ar − a`)

⋮f(bp+m−`−1)⋯f(bN−2)⋯bN−2f(ar − a`)

f(bp+m−`−1)⋯f(bN−1)

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

,

where µ` = b`⋯bp−1f(a1−a`)⋯f(a`−1−a`)

[f(a1−a`)⋯f(a`−1−aj)f(a`+1−a`)⋯f(ak−a`)]nf(a1−a`)⋯f(ar−a`).

(2) r < j ≤ kIn this case, α is in the last complete block. Here, we let p = k(n − 1) + `

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28 GONG, MEYERSON, MEZA, WARD, STOICIU

Now, we claim

u` = µ`

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

0⋮0

f(bp+m−`−1)f(a1 − a`)⋯(ar − a`)f(am+1 − a`)⋯f(ak − a`)f(bp+m−`−1)f(bp+m−l)f(a1 − a`)⋯f(ar − a`)f(am+2 − a`)⋯f(ak − a`)

⋮f(bp+m−`−1)⋯f(bp+k−`−1)f(a1 − a`)⋯f(ar − a`)f(bp+m−`−1)⋯f(bp+k−`)f(a2 − a`)⋯f(ar − a`)

⋮f(bp+m−`−1)⋯f(bN−2)f(ar − a`)

f(bp+m−`−1)⋯f(bN−1)

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

,

where

µ` =b`⋯bp−1f(a1 − a`)⋯f(a`−1 − a`)

[f(a1 − a`)⋯f(a`−1 − a`)f(a`+1 − aj)⋯f(ak − a`)]nf(a1 − a`)⋯f(ar − a`).

From these formulas, we can find the eigenvectors, and hence the asymptotic be-havior of the ε-pseudospectrum for any bidiagonal matrix,

σε(A) ≈k

⋃j=1

B (aj , (Cjε)1nj )

where Cj = ∥vj∥∥uj∥ and nj is the size of the Jordan block corresponding to aj .

Note 1: Let A be a periodic, bidiagonal matrix and suppose bi = 0 for somei. Then the matrix decouples into the direct sum of smaller matrices, call themA1, . . . ,An. To find the ε-pseudospectrum of A, apply the same analysis to thesesmaller matrices, and from Theorem 2.2, we have that

σε(A) =n

⋃i=1

σε(Ai).

Note 2: It is also interesting that given the results in Case 4, we can find differentmatrices with the same asymptotic ε-pseudospectra. Based on the formulas for theright and left eigenvectors, we observe that we can change the arrangement of theperiod without changing the left and right eigenvectors for any eigenvalue if thefollowing hold:

(1) The algebraic multiplicity of each ai remains the same.(2) The first position of each distinct eigenvalue remains the same. Addition-

ally, suppose α first appears in position `. We can permute the previousentries in positions 1,2, . . . , `−1 as long as we fix the number of times eachdistinct entry appears.

(3) The last position of each distinct eigenvalue remains the same in a period.Additionally, suppose α last appears in position q on the main diagonal. Letq ≡m (mod k). We can permute the other entries in positions m+ 1, . . . , kas long as we fix the number of times each distinct entry appears.

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EXPLICIT BOUNDS FOR THE PSEUDOSPECTRA OF MATRICES AND OPERATORS 29

Example 6.1. Let matrix A have period 13 on the main diagonal, with distincteigenvalues a, b, c. Let A have period

{a, b, a, b, c, a, b, c, a, b, c, b, c}.Note that the first entries of a, b, c are fixed in positions 1,2,5 respectively and thelast entries are fixed in positions 9,12,13 respectively.

We can permute entries 3,4 since permuting these entries does not change the num-ber of a or b entries before the first entry of c. We can also permute entries 6,7,8,as well as the entries 10,11 for similar reasons. We now give possible permutationsof said entries:

● A1 has period {a, b, b, a, c, a, b, c, a, b, c, b, c}, permuting entries 3 and 4,● A2 has period {a, b, a, b, c, b, c, a, a, b, c, b, c}, permuting entries 6, 7, 8,● A3 has period {a, b, b, a, c, a, b, c, a, c, b, b, c}, permuting entries 10 and 11.

With these possible permutations,

σε(A) ≈ σε(A1) ≈ σε(A2) ≈ σε(A3)

We will now present examples of bidiagonal matrices with explicit and asymptoticbounds.

6.2. Examples of Bidiagonal Matrices.

Example 6.2. Let A be an n × n bidiagonal matrix,

A =

⎛⎜⎜⎜⎜⎜⎝

a b1 0 0 . . .0 a b2 0 . . .⋮ ⋮ ⋱ ⋱ . . .⋮ ⋮ ⋮ ⋱ bn−10 0 0 0 a

⎞⎟⎟⎟⎟⎟⎠

where bi ≠ 0. The following propositions give explicit lower and upper bounds forthe ε-pseudospectrum of A.

Proposition 6.1 (Lower Bound).

B (a, (ε∣b1b2⋯bn−1∣)1n ) ⊆ σε(A).

Proof. Let z ∈ B (a, (ε∣b1b2⋯bn−1∣)1n ). We will show that z ∈ σε(A). Let E be an

n × n matrix with ε in the lower-left corner and 0’s elsewhere. Then ∥E∥ ≤ ε. Wetake the determinant of A + E − z and set it equal to 0 to get the eigenvalues ofA+E.

0 = ∣A +E − z∣ = (λ − z)n + (−1)n−1εb1b2⋯bn−1Ô⇒ (λ − z)n = (−1)nεb1b2⋯bn−1

Ô⇒ z = λ + (εb1b2⋯bn−1)1n .

Thus, z ∈ σε(A). �

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30 GONG, MEYERSON, MEZA, WARD, STOICIU

Proposition 6.2 (Upper Bound). The Jordan decomposition of A is

A =

⎛⎜⎜⎜⎜⎜⎜⎝

11b1

1b1b2

⋱1

b1⋯bn−1

⎞⎟⎟⎟⎟⎟⎟⎠

⎛⎜⎜⎜⎜⎜⎝

a 1a 1

a ⋱⋱ 1

a

⎞⎟⎟⎟⎟⎟⎠

⎛⎜⎜⎜⎜⎜⎝

1b1

b1b2⋱

b1⋯bn−1

⎞⎟⎟⎟⎟⎟⎠

.

Thus σε(A) ⊆ B (a, (∣b1⋯bn−1∣ε)1n + o (ε1/n)).

Combining this upper bound with the lower bound from Prop. 6.1, we get that

asymptotically, σε(A) ≈ B (a, (∣b1⋯bn−1∣ε)1n ).

Note that if bi = 0 for some i, then the matrix decouples, such that its Jordan

decomposition has multiple blocks and σε(A) ⊆ B (a, (∣bk⋯bj ∣ε)1n + o (ε1/n)) where

bk, bk+1, . . . bj correspond to the largest Jordan block.

We now find bounds for a bidiagonal matrix of period 2 on the main diagonal.

Example 6.3. Restricting to period 2 on the main diagonal and constants on thesuperdiagonal, we can get explicit lower bounds for our pseudospectra.

Let A be a 2N × 2N bidiagonal matrix with period 2 on the main diagonal, suchthat

A =

⎛⎜⎜⎜⎜⎜⎝

a1 1a2 1

⋱ ⋱a1 1

a2

⎞⎟⎟⎟⎟⎟⎠

.

Proposition 6.3 (Lower Bound).

2

⋃j=1

B (aj ,1

∣a1 − a2∣(ε

1N )) ⊆ σε(A).

Proof. Fix θ ∈ [0,2Nπ), and let E be the perturbation matrix with εeiθ on thelower left corner.

Then, det(A +E − z) = 0⇒ (a1 − z)N(a2 − z)N − ε = 0. Solving for z, we find that

z =a1 + a2 ±

√(a1 − a2)2 + 4(εeiθ)1/N

2.

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EXPLICIT BOUNDS FOR THE PSEUDOSPECTRA OF MATRICES AND OPERATORS 31

Around the eigenvalue λ1 as ε→ 0, we have

z =a1 + a2 +

√(a1 − a2)2 + 4(εeiθ)1/N

2

= a1 + a22

+ a1 − a22

⎡⎢⎢⎢⎢⎢⎣1 +

4 (εeiθ)1N

(a1 − a2)2

⎤⎥⎥⎥⎥⎥⎦

1/2

≈ a1 + a22

+ a1 − a22

⎡⎢⎢⎢⎢⎢⎣1 +

2 (εeiθ)1N

(a1 − a2)2

⎤⎥⎥⎥⎥⎥⎦

= a1 +(εeiθ)1/N

a1 − a2.

So, ∣z − a1∣ ≈ ε1N

∣a1−a2∣eiθ/N .

Performing the same calculation for the eigenvalue a2, we find that

∣z − a2∣ ≈ε

1N

∣a1 − a2∣eiθ/N .

Ranging θ over [0,2Nπ), we get that the disks of radius ε1N

∣a1−a2∣, centered around

a1 and a2 are contained in our ε-pseudospectrum. In other words,

2

⋃j=1

B (aj ,1

∣a1 − a2∣(ε

1N )) ⊆ σε(A).

We also have an asymptotic upper bound for σε(A):

∣z − aj ∣ ≤ε

1N

∣a1 − a2∣(1 + ∣a1 − a2∣2)

12N

and the ratio of the upper and lower bounds is just (1 + ∣a1 − a2∣2)1

2N .

Proposition 6.4 (Asymptotic Lower Bound).

2

⋃j=1

B (aj ,1

∣a1 − a2∣(√

1 + ∣a1 − a2∣2ε)1N + o(ε

1N )) ⊆ σε(A).

Proof. From Theorem 4.1, we can find an asymptotic lower bound for the bidiagonalmatrix of period 2.

We Jordan decompose matrix A such that QAP = J where QP = I. Then, wemultiply the first column of P , call it x1, by the N th row of Q, call it y1 to get theconstant that multiplies a1. We find that

∥x1y1∥ =1

∣a1 − a2∣N√

1 + ∣a1 − a2∣2.

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32 GONG, MEYERSON, MEZA, WARD, STOICIU

For a2, we find x2y2 by multiplying the N th column of P with the last row of Q,and get that

∥x2y2∥ = ∥x1y1∥.So, as ε→ 0

2

⋃j=1

B (aj ,1

∣a1 − a2∣(√

1 + ∣a1 − a2∣2ε)1N + o(ε

1N )) ⊆ σε(A)

This lower bound differs from the lower bound given in Proposition 6.3 by a con-

stant, α = (√

1 + ∣a1 − a2∣2)1N and some o(ε 1

N ).

7. Tridiagonal Matrices

The natural extension to bidiagonal matrices is to consider tridiagonal matrices.The study of tridiagonal matrices arises from a quantum mechanics random matrixproblem known as the Anderson model, as introduced by Anderson [1]. Hatano,Nelson, and Shnerb [?] studied the non-Hermitian analog of the Anderson model,now known as the Hatano-Nelson model, which takes the form

A =

⎛⎜⎜⎜⎜⎜⎜⎜⎝

a1 eg e−g

e−g a2 eg

⋱ ⋱ ⋱e−g aN−1 eg

eg e−g aN

⎞⎟⎟⎟⎟⎟⎟⎟⎠

where g is a fixed real parameter and a1, . . . aN are iid random variables. We areinterested in the nonperiodic Hatano-Nelson matrix

A =

⎛⎜⎜⎜⎜⎜⎜⎜⎝

a1 eg

e−g a2 eg

⋱ ⋱ ⋱e−g aN−1 eg

e−g aN

⎞⎟⎟⎟⎟⎟⎟⎟⎠

In this case, we can use a diagonal similarity transformation ofD = diag(1, eg, e2g, . . . , e(N−1)g)to get that the nonperiodic Hatano-Nelson matrix is similar to the matrix

DAD−1 =

⎛⎜⎜⎜⎜⎜⎜⎜⎝

a1 11 a2 1

⋱ ⋱ ⋱1 aN−1 1

1 aN

⎞⎟⎟⎟⎟⎟⎟⎟⎠

This matrix will only have one Jordan block per eigenvalue [7]. In fact, since thismatrix is symmetric and tridiagonal with nonzero off-diagonal entries, then we knowit has N distinct eigenvalues [12, 6].

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EXPLICIT BOUNDS FOR THE PSEUDOSPECTRA OF MATRICES AND OPERATORS 33

For the case when a = a1 = a2 = . . . = aN , then A is Toeplitz, and it is known thatthe eigenvalues are of the form λk = a + 2 cos ( kπ

N+1) and eigenvectors take the form

u(k)j = sin kjπ

n+1[6, 15]. The Toeplitz case has been studied [11] and the sensitivity

of the eigenvalues investigated; however by finding the eigenvectors, we can giveasymptotic results on the pseudospectrum.

For the case when the main diagonal is periodic with period 2, i.e.

aj =⎧⎪⎪⎨⎪⎪⎩

a1 if j odd

a2 if j even

then we can find an explicit formula for the eigenvalues and eigenvectors [9, 6].With these formulae, we can then find an asymptotic upper bound on σε(A), as wedid for bidiagonal matrices.

Further research would attempt to generalize these results to period k on the maindiagonal, and then ultimately to arbitrary diagonal entries.

8. Finite Rank operators

The majority of this paper has focused on both explicit and asymptotic characteri-zations of ε-pseudospectra for various classes of finite dimensional linear operators.A natural next step is to consider finite rank operators on infinite dimensionalspace.

In section 2 we defined ε-pseudospectra for matrices, although our definitions areexactly the same in the infinite dimensional case. For our purposes, the only note-worthy difference between matrices and operators is that the spectrum of an oper-ator is no longer defined as the collection of eigenvalues, but rather

σ(A) = {λ ∣ λI −A does not have a bounded inverse}As a result, we do not get the same properties for pseudospectra as we did previ-ously; in particular, σε(A) is not necessarily bounded.

That being said, the following theorem shows that finite rank operators behavesimilarly to matrices, in that asymptotically the radii ε-pseudospectra are boundedby roots of epsilon. The following theorem makes this precise.

Theorem 8.1. Let V be a Hilbert space and A ∶ V → V a finite rank operator onH. Then there exists C such that for sufficiently small ε,

σε(A) ⊆ σ(A) +B(0,Cε1

m+1 ).where m is the rank of A. Furthermore, this bound is sharp in the sense that thereexists a rank-m operator A and a constant c such that

σε(A) ⊇ σ(A) +B(0, cε1

m+1 )for sufficiently small ε.

Proof. Since A has finite rank, there exists a finite dimensional subspace U suchthat V = U ⊕W and A(U) ⊆ U , and A(W ) = {0}. Choosing an orthonormal basis

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34 GONG, MEYERSON, MEZA, WARD, STOICIU

for A which respects this decomposition we can write A = A′⊕0. Then the spectrumof A is σ(A′) ∪ {0}, and we know that for any ε,

σε(A) = σε(A′) ∪ σε(0).

The ε-pseudospectrum of the zero operator is well-understood since this operatoris normal; for any ε, it is precisely the ball of radius ε. It thus suffices to considerthe ε-pseudospectrum of the finite rank operator A′ ∶ U → U , where U is finitedimensional. The ε-pseudospectrum of this operator goes like ε1/j , where j is thedimension of the largest Jordan block; we will prove that j ≤m + 1. Note that therank of the n × n Jordan block given by

A =

⎛⎜⎜⎜⎜⎜⎝

λ 1 0 0 . . .0 λ 1 0 . . .⋮ ⋮ ⋱ ⋱ . . .⋮ ⋮ ⋮ ⋱ bn−10 0 0 0 λ

⎞⎟⎟⎟⎟⎟⎠

is n if λ ≠ 0, and n − 1 if λ = 0. Since we know that the rank of A is larger thanor equal to the rank of the largest Jordan block, we have an upper bound on thedimension of the largest Jordan block: it is of size m + 1, with equality attainedwhen λ = 0. By previous theorems, we then know that σε(A) is contained, for small

enough ε, in the set σ(A) +Cε 1m+1 .

Note that this bound is sharp; we can see this by taking V to be Rm+1 and consid-ering the rank-m operator

Am =

⎛⎜⎜⎜⎜⎜⎝

0 1 0 0 . . .0 0 1 0 . . .⋮ ⋮ ⋱ ⋱ . . .⋮ ⋮ ⋮ ⋱ 10 0 0 0 0

⎞⎟⎟⎟⎟⎟⎠

the pseudospectrum of which will contain the ball of radius ε1/m+1 by proposition5.1. �

The natural question to ask now is whether we can extend this result to morearbitrary operators on Hilbert spaces. For operators on Banach spaces, we dostill have a certain convergence of the ε-pseudospectrum to the spectrum [14, §4],namely ∩ε>0 σε(A) = σ(A). Also, while the ε-pseudospectrum may be unbounded,any bounded component of it necessarily contains a component of the spectrum.

From these results, we see that we can no longer get a complete asymptotic result forσε(A), as there could be an unbounded component of σε(A) that has no intersectionwith the spectrum. For example, there exist operators with bounded spectrum andunbounded ε-pseudospectra for all ε > 0.

However, the above result states that the bounded components of the ε-pseudospectrummust converge to the spectrum. If we restrict our attention to these bounded com-ponents, we can generalize Thms. 4.1 and 8.1 by asking whether the boundedcomponents of σε(A) converge to the spectrum as a union of disks.

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EXPLICIT BOUNDS FOR THE PSEUDOSPECTRA OF MATRICES AND OPERATORS 35

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