Numerical Range of Square Matrices : A Study in Spectral Theory

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(1)Numerical Range of Square Matrices A Study in Spectral Theory Department of Mathematics, Linköping University Erik Jonsson LiTH-MAT-EX–2019/03–SE. Credits: 16 hp Level: Supervisor:. G2 Göran Bergqvist, Department of Mathematics, Linköping University. Examiner: Milagros Izquierdo, Department of Mathematics, Linköping University Linköping:. June 2019.

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(3) Abstract In this thesis, we discuss important results for the numerical range of general square matrices. Especially, we examine analytically the numerical range of complex-valued 2 × 2 matrices. Also, we investigate and discuss the Gershgorin region of general square matrices. Lastly, we examine numerically the numerical range and Gershgorin regions for different types of square matrices, both contain the spectrum of the matrix, and compare these regions, using the calculation software Maple. Keywords: numerical range, square matrix, spectrum, Gershgorin regions URL for electronic version: urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-157661. Erik Jonsson, 2019.. iii.

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(5) Sammanfattning I denna uppsats diskuterar vi viktiga resultat för numerical range av generella kvadratiska matriser. Speciellt undersöker vi analytiskt numerical range av komplexvärda 2 × 2 matriser. Vi utreder och diskuterar också Gershgorin områden för generella kvadratiska matriser. Slutligen undersöker vi numeriskt numerical range och Gershgorin områden för olika typer av matriser, där båda innehåller matrisens spektrum, och jämför dessa områden genom att använda beräkningsprogrammet Maple. Nyckelord: numerical range, kvadratisk matris, spektrum, Gershgorin områden URL för elektronisk version: urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-157661. Erik Jonsson, 2019.. v.

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(7) Acknowledgements I’d like to thank my supervisor Göran Bergqvist, who has shown his great interest for linear algebra and inspired me for writing this thesis. Göran also stood up for giving me this very exciting subject, which can be researched in different levels, but making this possible for my bachelor thesis. I’d also like to thank my examiner Milagros Izquierdo, who has organized the oral presentation part where I’ve shown my work for my peer reviewers. I’d also like to thank my childhood friend Love Mattsson, who has asked interesting questions about the subject and giving thumbs up for my work. Lastly, I’d like to thank my family who has supported me and listened to my ideas about generalizing the subject to a level they’d understand and giving me some feedback to old versions of the thesis.. Erik Jonsson, 2019.. vii.

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(9) Nomenclature A - Square matrix A Mn - Set of all square matrices with dimension n A−1 - Inverse of A AT - Transpose of A A∗ - Hermitian transpose of A Γ - Schur form of A L Ai - Direct sum of square matrices in Mn λ - Eigenvalue of A σ(A) - Spectrum of A F(A) - Numerical range of A kAkF - Frobenius norm of A G (i) (A) - Gershgorin disc of A with index i G(A) - Gershgorin region of A. Erik Jonsson, 2019.. ix.

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(11) Contents List of Figures. xiii. 1 Introduction. 1. 2 Preliminaries 2.1 Spectral theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Fundamental concepts from Optimization and Multi-variable Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 3 3. 3 Numerical Range 3.1 Numerical range of general square matrices . . . 3.2 Numerical range of a 2 × 2 matrix . . . . . . . . 3.3 Topology behind the numerical range of a general 3.4 General results of normal matrices . . . . . . . .. . . . . . . . . . . matrix . . . . .. . . . .. . . . .. . . . .. . . . .. 4 Gershgorin Discs 5 Results 5.1 Theory behind the illustrations . . . . . . . . 5.1.1 Drawing the illustrations . . . . . . . 5.2 General matrices in M2 . . . . . . . . . . . . 5.3 Normal matrices . . . . . . . . . . . . . . . . 5.4 Almost normal matrices . . . . . . . . . . . . 5.4.1 Interference of interior points . . . . . 5.4.2 Interference of boundary points . . . . 5.5 Other matrices that are not normal . . . . . . 5.6 Gershgorin region versus Numerical range . . 5.6.1 Matrices in M2 . . . . . . . . . . . . . 5.6.2 Matrices in M3 and higher dimensions Erik Jonsson, 2019.. 7 9 9 12 17 18 21. . . . . . . . . . . .. . . . . . . . . . . .. . . . . . . . . . . .. . . . . . . . . . . .. . . . . . . . . . . .. . . . . . . . . . . .. . . . . . . . . . . .. . . . . . . . . . . .. . . . . . . . . . . .. . . . . . . . . . . .. . . . . . . . . . . .. 25 25 27 27 30 33 34 36 39 41 41 47 xi.

(12) xii. 6 Discussion 6.1 Numerical range . . . . . . . . . . . . . . . 6.2 Gershgorin regions . . . . . . . . . . . . . . 6.3 Numerical range versus Gershgorin regions . 6.4 Further topics to investigate . . . . . . . . . Bibliography. Contents. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. 61 61 62 62 62 65. A Maple Commands(Code) 67 A.1 Numerical range . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 A.2 Gershgorin region . . . . . . . . . . . . . . . . . . . . . . . . . . . 68.

(13) List of Figures 3.1 3.2. Numerical range of A in Example 3.5 Numerical range of A in Example 3.7. . . . . . . . . . . . . . . . . . . . . . . . .. 12 16. 4.1 4.2. Gershgorin region of A in Example 4.2 . . . . . . . . . . . . The intersection of G(A) and G(AT ) in Example 4.5 . . . .. 22 24. 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9 5.10 5.11 5.12 5.13 5.14 5.15 5.16 5.17 5.18 5.19 5.20 5.21 5.22 5.23. Numerical range of A1 . . . . . . . . . . . . . . . . . . Envelope for F(A1 ) . . . . . . . . . . . . . . . . . . . . Numerical range of A2 . . . . . . . . . . . . . . . . . . Envelope for F(A2 ) . . . . . . . . . . . . . . . . . . . . Numerical range of A3 . . . . . . . . . . . . . . . . . . Envelope for F(A3 ) . . . . . . . . . . . . . . . . . . . . Numerical range of A4 . . . . . . . . . . . . . . . . . . Envelope for F(A4 ) . . . . . . . . . . . . . . . . . . . . Numerical range of A5 . . . . . . . . . . . . . . . . . . Envelope for F(A5 ) . . . . . . . . . . . . . . . . . . . . Numerical range of A6 . . . . . . . . . . . . . . . . . . Envelope for F(A6 ) . . . . . . . . . . . . . . . . . . . . Numerical range of A7 . . . . . . . . . . . . . . . . . . Envelope for F(A7 ) . . . . . . . . . . . . . . . . . . . . Numerical range of A8 . . . . . . . . . . . . . . . . . . Envelope for F(A8 ) . . . . . . . . . . . . . . . . . . . . e8 ) when r = 6 . . . . . . . . . . . . . Envelope for F(A Envelope for F(Aˆ8 ) when r = 83 . . . . . . . . . . . . . Envelope for F(Aˆ8 ), the point λ2 = 2 + 3i is zoomed Envelope for F(Aˆ8 ) when r = 5 . . . . . . . . . . . . . Numerical range of A9 . . . . . . . . . . . . . . . . . . Envelope for F(A9 ) . . . . . . . . . . . . . . . . . . . . e9 . . . . . . . . . . . . . . . . . . Numerical range of A. 27 27 28 28 29 29 30 30 31 31 32 32 33 33 34 34 35 36 37 38 39 39 40. Erik Jonsson, 2019.. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . .. xiii.

(14) xiv. List of Figures. 5.24 5.25 5.26 5.27 5.28 5.29 5.30 5.31 5.32 5.33 5.34 5.35 5.36 5.37 5.38 5.39 5.40 5.41 5.42 5.43 5.44 5.45 5.46 5.47 5.48 5.49 5.50 5.51. e9 ) . . . . Envelope for F(A Envelope for F(A10 ) . . . . Gershgorin region for A10 Intersection of F(A10 ) and Envelope for F(A11 ) . . . . Gershgorin region for A11 Intersection of F(A11 ) and Envelope for F(A12 ) . . . . Gershgorin region for A12 Intersection of F(A12 ) and Envelope for F(A13 ) . . . . Gershgorin region for A13 Intersection of F(A13 ) and Envelope for F(A14 ) . . . . Gershgorin region for A14 Intersection of F(A14 ) and Envelope for F(A15 ) . . . . Gershgorin region for A15 Intersection of F(A15 ) and Envelope for F(A16 ) . . . . Gershgorin region for A16 Intersection of F(A16 ) and Envelope for F(A17 ) . . . . Gershgorin region for A17 Intersection of F(A17 ) and Envelope for F(A18 ) . . . . Gershgorin region for A18 Intersection of F(A18 ) and. . . . . . . . . . . . . . . . G(A10 ) . . . . . . . . . . G(A11 ) . . . . . . . . . . G(A12 ) . . . . . . . . . . G(A13 ) . . . . . . . . . . G(A14 ) . . . . . . . . . . G(A15 ) . . . . . . . . . . G(A16 ) . . . . . . . . . . G(A17 ) . . . . . . . . . . G(A18 ). . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 40 41 41 42 43 43 44 45 45 46 47 47 48 49 49 50 51 51 52 53 53 54 56 56 57 59 59 60.

(15) Chapter 1. Introduction In technical applications such as dynamical systems, spectral theory is of huge interest. It’s often that systematic properties such as stability for a mechanical system are analyzed with the help of the study of eigenvalues of a linear system. Spectral theory is also useful in mathematical areas such as optimization w.r.t. no constraints, where a matrix A ∈ Mn , is analyzed to determine if it’s definite or not, depending on the form of the function that describes the matrix. Studying eigenvalues of probabilistic matrices (called Markovian matrices) that are positive, is also of huge interest in network problems (where a network is a graph). In this thesis, we are interested in investigating how the set of complex-valued numbers that forms the spectrum of A, σ(A), relate to its matrix geometrically, in a number of cases. As we find the results, describing the comparison of numerical range and Gershgorin region of a square matrix containing σ(A) is essential to conclude which of them to prefer.. Erik Jonsson, 2019.. 1.

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(17) Chapter 2. Preliminaries Before we mention the properties of the numerical range of arbitrary square matrices, let’s refresh our memories regarding fundamental definitions in spectral theory for general square matrices, and of compact and convex sets. We also remind ourselves about the definition of an envelope and Frobenius norm.. 2.1. Spectral theory. Recall the following definition: Definition 2.1 (Definition 8.1.1, Janfalk [4]) Let V be a vector space and consider the linear map T : V → V . A number λ ∈ C is called the eigenvalue of T if it exists a non-zero vector u ¯ ∈ V s.t. T (¯ u) = λ¯ u. The vector u ¯ above is called the eigenvector and corresponds to the eigenvalue λ of T . In this thesis, the interest lies in studying matrices A ∈ Mn , which maps Cn → Cn . Then ¯ 0 6= u ¯ ∈ Cn , is an eigenvector to A, with eigenvalue λ ∈ C, if A¯ u = λ¯ u. It’s easy to determine the eigenvectors that corresponds to the eigenvalues of A. To refresh our memories, let’s do an example.. Erik Jonsson, 2019.. 3.

(18) 4. Chapter 2. Preliminaries. 4 9 . Calculations by 9 4 using the definition of secular polynomial for A gives us Example 2.2 Consider the following matrix A =. λ1 = −5, λ2 = 13 which corresponds to the eigenvectors of A u ¯1 = (−1, 1), u ¯2 = (1, 1) respectively. It’s easy to check that A¯ u1 = λ1 u ¯1 , A¯ u2 = λ2 u ¯2 . If it exist a basis of eigenvectors [¯ u1 , u ¯2 , . . . , u ¯n ] that corresponds to the eigenvalues of A, λ1 , λ2 , ..., λn , then A = T DT −1 , where T has u ¯1 , u ¯2 , . . . , u ¯n as columns. We say A can be diagonalized with a diagonal matrix   λ1 . . . 0 0    0 λ2 . . . 0   D= . . . .. . . . ..  .  ..  0 0 . . . λn However, there are many cases when A cannot be diagonalized. During these situations, we can use, for instance, a Jordan block in canonical form that makes A almost diagonalizable. Recall the following definitions of Hermitian, normal, unitary and orthogonal matrices. Definition 2.3 (Treil [9]) A∗ is defined as the transpose and complex conjugate of A. Definition 2.4 (Treil [9]) A is Hermitian if A = A∗ . Definition 2.5 (Treil [9]) A is normal if AA∗ = A∗ A. Definition 2.6 (Treil [9]) A is unitary if A∗ = A−1 . Definition 2.7 (Treil [9]) A is orthogonal if A is real and AT = A−1 ..

(19) 2.1. Spectral theory. 5. These definitions are important for diagonalizing square matrices, depending on whether the matrices are normal or not. Recall the following theorems of Schur factorization of a matrix A and diagonalization of normal matrices. Theorem 2.8 Let A be a matrix (or map A : X → X acting on a complex vector space). Then there exists an ON-basis {¯ ui } s.t. the matrix Γ of A becomes upper triangular. In other words, A can be represented as A = U ΓU ∗ , where U is unitary. Proof: See proof in [9, Chapter 6]. Theorem 2.9 Spectral theorem for normal matrices Let A be a normal square matrix, AA∗ = A∗ A. Then A can be represented as A = U DU ∗ , where U is a unitary matrix and D diagonal matrix. Also, if A = A∗ is selfadjoint, then D is real-valued. Furthermore, if A is real symmetric, then U is an orthogonal matrix. Proof: See proof in [9, Chapter 6]. It’s important that the definition of the Frobenius norm of a matrix is mentioned too. Definition 2.10 Let A be an m × n matrix. Then the Frobenius norm is the matrix norm defined as [12] v uX n

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(21) 2 um X

(22) aij

(23) . kAkF = t i=1 j=1. The norm can also be written as following p kAkF = tr(A∗ A). We formulate also the definition of principal submatrix of a square matrix below..

(24) 6. Chapter 2. Preliminaries. Definition 2.11 Let A be a square matrix. A k × k principal submatrix of A ∈ Mn has elements aij , where i, j ∈ I (the same I for i and j) for some I = {i1 , . . . , ik : 1 ≤ i1 < i2 < . . . < ik ≤ n}. To understand how to obtain the principal submatrices of A, let’s do an example.   2 5 0 Example 2.12 Consider the matrix A = 1 3 3. If I1 = {1} and I2 = 0 1 4 {2, 3}, then the following principal submatrices of A that corresponds to I1 and I2 are 3 2 , 1. 3 4. . respectively. Note that if I3 = {1, 2, 3}, then the principal submatrix of A must be the matrix itself. Now we define the direct sum of matrices. L. .. Definition 2.13 Let Ai ∈ Mn , 1 ≤ i ≤ n. Then the direct sum of matrices constructs the block diagonal matrix from a set of square matrices [13]  A1  n M  Ai = diag(A1 , . . . , An ) =    i=1.  A2 ... . An.   .  . We make use of this definition for doing an example below. 1 3 2 4 , A2 = . Then the direct sum of A1 and Example 2.14 Let A1 = 5 8 7 9 A2 is the following block diagonal matrix  1 M 5 A1 A2 = diag(A1 , A2 ) =  0 0. 3 8 0 0. 0 0 2 7.  0 0 . 4 9.

(25) 2.2. Fundamental concepts from Optimization and Multi-variable Calculus. 2.2. 7. Fundamental concepts from Optimization and Multi-variable Calculus. Before we define concepts like convexity and compactness of the numerical range for arbitrary n × n matrices, let’s discuss about what it actually means to have a convex and compact set in Rn . Let’s begin with compactness. Definition 2.15 A set in Rn is compact if it’s both closed and bounded. [1] In Chapter 3 and 4, we’ll notice that the numerical range and Gershgorin regions of a square matrix, are compact subsets in C. Observe that the set (x, y) ∈ R2 : 3x2 + 4y 2 ≤ 19, 0 ≤ y ≤ x is closed and bounded. Now, recall the definition of convexity for a set in Rn . ¯1 , x ¯2 ∈ X Definition 2.16 A set X ⊆ Rn is a convex set if for any pair of points x and for 0 ≤ t ≤ 1, we have [5] x ¯ = t¯ x1 + (1 − t)¯ x2 ∈ X. The definition says that if a set is to be convex, then for every two points in X, the line segment joining them must be within that set. There are lots of examples when a certain set isn’t convex. Finally, recall the definition of an envelope. Definition 2.17 The envelope of a one-parameter family of curves, defined by [11] F (x, y, c) = 0, where c a parameter, is a curve that intersects every member of the family by their tangents. By the definition above, for finding the envelope of such a one-parameter family of curves, it suffices to solve the typical first-order differentiation equation ∂F = 0. ∂c F (x, y, c) = 0.

(26) 8. Chapter 2. Preliminaries. Later in this thesis, we’ll see that the boundary of the numerical range of a complex-valued n × n matrix, is the envelope of a family of lines that intersects the vertices of an oval region in xy-plane (the region bounded by an ellipse if the matrix is in M2 )..

(27) Chapter 3. Numerical Range In this chapter, we describe and formulate the concept numerical range for general matrices in Mn . Also, we define and show the concepts convexity, compactness and some properties for the numerical range of general square matrices. Having done this, we’ll be ready for examining Gershgorin regions of general square matrices (see Chapter 4) and later on, performing experiments on several matrices (see Chapter 5). A general reference for this chapter is Shapiro [8].. 3.1. Numerical range of general square matrices. Definition 3.1 The set of all complex numbers x ¯∗ A¯ x, where the norm k¯ xk = 1, is called the numerical range of A, and is denoted F(A). In other words, [8] ∗. F(A) = x ¯ A¯ x|¯ x ∈ Cn , k¯ xk = 1 . First a theorem that states useful properties about numerical range of general square matrices. Theorem 3.2 Let A be an n × n complex-valued matrix. Then the following holds, [8] (a) For any unitary matrix U , then F(A) = F(U ∗ AU ). (b) F(A) contains all of the eigenvalues of A. (c) F(A) contains all of the diagonal entries of A. (d) If Ak is any principal submatrix of A, then F(Ak ) ⊆ F(A). (e) If P is a n × k matrix with orthonormal columns, then F(P ∗ AP ) ⊆ F(A). Erik Jonsson, 2019.. 9.

(28) 10. Chapter 3. Numerical Range. Some of these properties are perhaps trivial to see if we compare with earlier results from fundamental spectral theory, but let’s prove these statements. Proof: We prove these properties in order. First, in order to prove (a), consider a unitary matrix U and the compact set B which is the unit sphere in Cn , and also put y¯ = Ux ¯. Since U is unique and gives a bijection B onto itself, we of have F(A) = y¯∗ A¯ y |¯ y ∈ B = y¯∗ A¯ y |¯ y = Ux ¯, x ¯∈B = x ¯∗ U ∗ AU x ¯|¯ x∈B = F(U ∗ AU ). To prove (b), let λ be an eigenvalue of A, and also let x ¯ be a corresponding normalized eigenvector. Then it follows that x ¯∗ Ax = λ¯ x∗ x ¯ = λ. This means λ ∈ F(A) and σ(A) ⊂ F(A). For (c), it follows from e¯∗i A¯ ei = aii , where e¯i is a vector in the standard basis of Cn , 1 ≤ i ≤ n. For (d), choose a permutation matrix Q s.t. Ak is formed from the first k rows and columns of Q∗ AQ (Q is unitary). From (a), we have that F(Q∗ AQ) = F(A). Consider now z¯ ∈ Ck with normk¯ z k = 1. Therefore, put z¯ˆ = (z1 , . . . , zk , 0, . . . , 0). ¯ We have now zˆ = 1 and z¯∗ Ak z¯ = z¯ ˆ∗ Q∗ AQz¯ˆ, so F(Ak ) ⊆ F(Q∗ AQ) = F(A). For the last part of the proof, (e), we prove in a similar way as previous part. Let z¯ ∈ Ck with norm k¯ z k = 1. Since P has columns that are orthonormal, P z¯ ∈ Cn , has the norm kP z¯k = 1. Now, z¯∗ P ∗ AP z¯ = (P z¯)∗ A(P z¯) ∈ F(A), and hence F(P ∗ AP ) is a subset of F(A). To make sense of this theorem, let’s do an example. 5 Example 3.3 Consider the real symmetric matrix A = 6 trix has the eigenvalues. 6 . This ma10. λ1 = 1, λ2 = 14. This means that the spectrum of A is σ(A) = {1, 14} (ordered from smallest to largest eigenvalue). Using the recently stated theorem, if we check (b), we have σ(A) ⊂ F(A), where F(A) = [1, 14]. Observe that if we also check the property (c), then we have a11 = 5 ∈ F(A), a22 = 10 ∈ F(A). Also, since A is a normal matrix, then we can represent it as A = U DU ∗ , where D is the diagonal matrix.

(29) 3.1. Numerical range of general square matrices. 11. and U is unitary. Calculating U and D, gives us 1 −3 √ U= 13 2. 2 1 ,D = 3 0. 0 . 14. This means by (a), F(A) = F(U ∗ AU ) = F(D). There are interesting behavior in special types of normal matrices. For instance, Hermitian matrices have their numerical range as the closed and bounded interval of all real-valued numbers with λ1 as the smallest and λn as the largest eigenvalue as endpoints. Also, we saw in the proof of the theorem above that the numerical range of a matrix A is the same as the numerical range of the upper triangular matrix Γ. In the normal case, Γ is the diagonal matrix D. We’ll use this fact to prove an important theorem below. Theorem 3.4 Let A ∈ Mn be normal, with eigenvalues λ1 , . . . , λn . Then F(A) is the closed convex hull of the points λ1 , . . . , λn . [8] Before we prove this theorem, recall the following definition of convex hull of a set of points X: If X = {λ1 , λ2 , . . . , λn }, then the convex hull of X is Conv(X) = {Σni=1 αi λi , Σni=1 αi = 1, αi ≥ 0} . This is the smallest convex set that contains the set of points X. Proof: First, consider the unitary matrix U s.t. U ∗ AU = D, where D is the diagonal matrix. By Theoremo3.2, F(A) = F(D). Now, we have nP Pn n 2 2 F(D) = i=1 |xi | λi | i=1 |xi | = 1 , which is the set of all convex combinations of λ1 , . . . , λn . To make use of the recently stated theorem, we do an example.   0 1 0 0 0 0 0 1  Example 3.5 Consider the normal matrix A =  0 0 1 0 . Calculating the 1 0 0 0 eigenvalues gives us the following diagonal matrix D = U ∗ AU.

(30) 12.  1 0  = 0 0. Chapter 3. Numerical Range. 0 1 0 0. 0 0 1 −2 + 0. √ i 3 2. 0 0 0 − 21 −. . √ i 3 2.   . . Now, with the fact from the proof of the previous theorem with F(A) = F(D), illustrating this in Maple gives us. Figure 3.1: Numerical range of A in Example 3.5. 3.2. Numerical range of a 2 × 2 matrix. The theorem below is perhaps one of the most important ones for this thesis, because the proof of it describes well how any 2 × 2 matrix has the numerical range as all complex-valued numbers in the region bounded by an ellipse. With the understanding of this theorem, we can interpret the numerical range of other square matrices in higher dimensions since they are all oval. Theorem 3.6 Let A be q the 2 × 2 complex-valued matrix with eigenvalues 2 2 2 λ1 and λ2 . Also, let r = kAkF −|λ1 | −|λ2 | , where kAkF is the Frobenius norm of A. Then F(A) consists of all points in the region bounded by the ellipse with foci λ1 and λ2 and minor axis of length r. [8].

(31) 3.2. Numerical range of a 2 × 2 matrix. 13. Proof: We show first that we only need to study matrices of the form C = t r , where r and t are positive real numbers. 0 −t ∗ If A has eigenvalues λ1 and λ2 , let A = U ΓU be the Schur factorization λ1 b of A, so Γ = , and U is the unitary matrix. Then F(A) = F(Γ) and 0 λ2 2 2 2 2 2 kAkF = kΓkF = |λ1 | +|λ2 | +|b| .. Now, let λ = " Γ=. λ1 +λ2 . 2. λ1 −λ2 2. 0. Then, we have # " λ1 +λ2 b 2 + λ2 −λ1 0 2. 0 λ2 +λ1 2. #. λ = 0. λ1 + λ2 b I, + −λ 2. where I is the identity matrix. With λ = teiθ , b = reiφ = reiθ ei(φ−θ) = reiθ eiα , we have λ1 + λ2 λ1 + λ2 0 t r 1 0 reiα iθ 1 iθ t I=e + I= Γ=e + 0 e−iα 0 −t 0 eiα 0 −t 2 2 λ1 + λ2 t r = V (eiθ + I)V ∗ , 0 −t 2 1 0 where V is and unitary. 0 e−iα Now, from the start of the proof, we have F(A) = F(Γ) = F(B), since t r t r 2 Γ = V BV ∗ where B = eiθ + λ1 +λ I. With C = , then 2 0 −t 0 −t 2 F(A) = eiθ F(C) + λ1 +λ , so F(A) is aquired from F(C) by two operations in 2 Euclidean geometry: rotation and translation. n We can understand this furtheroby 2 looking at the following set notation F(A) = eiθ z + λ1 +λ ∈ C | z ∈ F(C) . 2 We now study the matrix C. For any normalized vector x ¯, we have (eiθ x ¯)∗ C(eiθ x ¯) = ∗ x ¯ Cx ¯. This means that we only need to consider vectors whose first component " # cos φ2 x1 is real. Therefore, for the norm k¯ xk = 1, we assume x ¯ = iθ . Calcue sin φ2 lating the product x ¯∗ C x ¯ gives us " # h i t r cos φ φ φ ∗ −iθ 2 x ¯ A¯ x = cos 2 e = t cos2 φ2 + reiθ sin φ2 cos φ2 − sin 2 0 −t eiθ sin φ2.

(32) 14. Chapter 3. Numerical Range. t sin2 φ2 = t cos φ + 2r eiθ sin φ, where trigonometric identities as cos2 φ2 − sin2 φ2 = cos φ and 2 sin φ2 cos φ2 = sin φ have been used. So, we see that F(C) is the set of all numbers on the form tcos φ + 2r eiθ sin φ,. (3.1). where real parameters φ, θ ∈ [0, 2π]. We see that for t = 0, we have the set of all numbers 2r eiθ sin φ, which is the compact region of a disk with radius 2r centered at origin. We also get the boundary circle when sin φ = 1 and interior points when |sin φ| < 1. Therefore, consider the case t 6= 0. Identifying the real and imaginary part of (3.1), we have X = t cos φ + Y =. r cos θ sin φ 2. r sin θ cos φ 2. From this, we use spherical coordinates and shear mapping on space in R3 for showing that the set of such all points is the ellipse. Of course, the spherical coordinates are x = ρ cos θ sin φ y = ρ sin θ sin φ z = ρ cos φ. Now, if ρ = 2r and let θ ∈ [0, 2π], φ ∈ [0, π]. What we get, is the surface of the sphere of radius ρ centered at the origin. Let B denote the surface. From this, we have cos φ =. z 2z = . ρ r. Now, comparing the equations above, we get 2t z+x r Y = y.. X=. Now we’d like to do the shear map as mentioned earlier:     x x + 2t z r   T y  =  y  . z z.

(33) 3.2. Numerical range of a 2 × 2 matrix. 15. With this linear transformation, it maps B onto the surface of an ellipsoid, denoted E. Geometrically, the vertical projection of E onto the xy-plane is the filled ellipse. The map T only shears in the x-direction and the shear depends only on the z-coordinate. Because of this, we have a symmetric ellipsoid obtained (from the sphere) with a shear in the xz-plane with its axis being the z-axis. The projected ellipse has its axes along the x - and y-coordinates. Moreover, the vertices on y-axis are the points (0, ± 2r ). To find the vertices on x-axis, let’s find the values of X. We have r cos φ t t cos φ + cos θ sin φ = · r . sin φ 2 2 cos θ. Since the norm (cos φ, sin φ) = 1, with Cauchy-Schwarz inequality theorem, we get

(34) r

(35) 2

(36)

(37)

(38) t cos φ + r cos θ sin φ

(39) ≤ t2 + r cos2 θ.

(40)

(41) 2 4 r Therefore, we choose θ = 0 and φ in a way that tan φ = 2t . The vectors in the dot product above becomes parallel and with equality from inequality equation above, we get

(42) r

(43) 2

(44)

(45) r

(46) t cos φ + sin φ

(47) = t2 + r .

(48)

(49) 2 4. Finally, the vertices on x-axis are r (± t2 +. r2 , 0). 4. We conclude lastly that the major axis of the ellipse is on the x-axis, the foci are at (±t, 0) and the minor axis has length r. 2 This ellipse is F(C) and with F(A) = eiθ F(C) + λ1 +λ , F(A) is the ellipse 2 λ1 +λ2 iθ obtained from rotation e and translation 2 of F(C). . We’ll do an example of how the theorem works. . 3 4 . This matrix has 1 6 the eigenvalues λ1 = 2, λ2 = 7. Now, there exist an upper triangular matrix Γ such that A = U ΓU ∗ , where U is unitary. Calculations by using Gram-Schmidt Example 3.7 Consider the real-valued matrix A =.

(50) 16. Chapter 3. Numerical Range. 7 3 . From the proof of previous theorem, 0 2 we calculate the parameters r and t to get a sense of how the numerical range of A looks. Since we know the formula of r, then we get r = 3 by using the definition of Frobenius norm of A. Also, from Γ, we see that θ = 0, t = 52 and λ = 92 . Therefore, the centre is ( 92 , 0), semi minor axis ± 32 , the semi major axis q √ √ 2 ± 34 + ( 52 )2 = ± 234 and the vertices are at 92 ± 234 . This means that the ellipse has λ1 and λ2 as foci (also interior points of the ellipse). process for U gives us that Γ =. Illustrating this in Maple, we get. Figure 3.2: Numerical range of A in Example 3.7. Now that we’ve stated some important theorems, let’s define the topological properties convexity and compactness of the numerical range of a square matrix of dimension n..

(51) 3.3. Topology behind the numerical range of a general matrix. 3.3. 17. Topology behind the numerical range of a general matrix. We now describe topological properties of the numerical range as a subset of the complex plane C. Theorem 3.8 Let F(A) be the numerical range of any matrix A ∈ Mn . Then, the following properties holds for F(A): [3] (a) F(A) is a compact subset of C. (b) F(A) is a convex subset of C. Proof: Let’s prove (a) first. F(A) is the range of a continuous map x ¯→x ¯∗ A¯ x k n ∗ : from C → C over the domain {¯ x x ¯ ∈ C ,x ¯ x ¯ = 1}, which is the unit sphere in Cn . Since the continuous image of a compact set is compact, then it follows F(A) is a compact set (we identify Cn with R2n when defining compact sets). For proving (b), we use the fact in 2 × 2 matrix case that it’s convex, and prove for the general n × n matrix case. Let two numbers a,b ∈ F(A). We need to show that any point on the line segment joining a and b is in F(A). It’s trivial if a = b, so consider a 6= b. Now let x ¯ and y¯ be two normalized vectors s.t. a=x ¯∗ A¯ x and b = y¯∗ A¯ y . Since we know that a 6= b, x ¯ and y¯ must be linearly independent. Let V be a subspace that is spanned by x ¯ and y¯ and also let v¯1 and v¯2 be a ON-basis for V . Then there are scalar numbers ci , dj , i, j = 1, 2 s.t. x ¯ = c1 v¯1 + c2 v¯2 y¯ = d1 v¯1 + d2 v¯2 c1 d and d¯ = 1 . Since we know that v¯1 and v¯2 are orthonormal c2 d2 2 2 2 2 and the norms k¯ x k = 1 and k¯ y k = 1, then |c1 | + |c2 | = |d1 | + |d2 | = 1, so ¯ k¯ ck = d = 1. Finally, let P be the n × 2 matrix with columns v¯1 and v¯2 , Write c¯ =. ¯ From this, we have a = x then x ¯ = P c¯ and y¯ = P d. ¯∗ A¯ x = c¯∗ P ∗ AP c¯ and ∗ ∗ ∗ ¯ ¯ b = y¯ A¯ y = d P AP d, which gives a and b are points in F(P ∗ AP ). However, ∗ P AP is a 2 × 2 matrix, so F(P ∗ AP ) contains the line segment with endpoints a and b. We also know that F(P ∗ AP ) ⊆ F(A), then it means F(A) contains the very same line segment with endpoints a and b. We’ll now discuss general results of normal square matrices..

(52) 18. 3.4. Chapter 3. Numerical Range. General results of normal matrices. Before we head on to the next chapter about Gershgorin regions, we’ll discuss the general results when we have a typical normal square matrix. Just before we discuss the results for a normal matrix, let’s prove the following theorem. Theorem 3.9 Let A ∈ Mn . If any eigenvalue λ of AL is on the boundary of F(A), there exist a unitary matrix U s.t. U ∗ AU = λ A2 , where A2 is a matrix of order n − 1. [8] Proof: Let us suppose that λ is on the boundary of F(A). In that case, We choose corresponding normalized eigenvector u ¯1 and u ¯2 , . . . , u ¯n s.t. u ¯1 , u ¯2 , . . . , u ¯n becomes an ON-basis for Cn . Now, let U be a unitary matrix with columns u ¯1 , . . . , u ¯n . Then we have   λ a12 a13 . . . a1n   0  , U ∗ AU =  . .  A2 .  0 where A2 is a (n−1)×(n−1) block matrix. Since F(A) = F(U ∗ AU ), so λ lies on the boundary of the numerical range of the matrix above. By this construction, λ a1j we show that a1j = 0, ∀j = 2, . . . , n. We build the matrix Cj = , a 2×2 0 aj j ∗ principal submatrix of U AU formed from rows and columns 1 and j. Obviously, we have F(Cj ) ⊆ F(A). If a1j 6= 0, then we have that F(Cj ) is a non-degenerate ellipse and λ is one of the foci, and therefore is an interior point of F(Cj ). This means of course that λ is an interior point of F(A). This is a contradiction because λ lies on the boundary of F(A). Because of that, a1j = 0, ∀j = 2, . . . , n. With this result, we can prove the following interesting theorem. Theorem 3.10 If A ∈ Mn with n ≤ 4, and F(A) is the convex hull of the eigenvalues of A, then A is a normal matrix. [8] In the upcoming proof, we use the knowledge about different types of numerical range to conclude if A is normal or not..

(53) 3.4. General results of normal matrices. 19. Proof: We make the assumption that F(A) is the convex hull of the eigenvalues of A. If A is a matrix with dimension n ≤ 4, there can only be at most four distinct eigenvalues, so F(A) is either a point, line segment, triangle or a quadrilateral. In the case that F(A) is a triangle or quadrilateral, we realize that the eigenvalues are vertices. If we consider a line segment as a subset in the plane, then every point joining the line segment obviously is a boundary point. Therefore, we realize in all cases that at least n − 1 of the n eigenvalues are on the boundary of F(A). There is an exception though, and that’s when A is a 4 × 4 matrix and its numerical range is a triangle, then the fourth eigenvalue is an interior point of the triangle. Observing these cases, we realize that A is normal. By using Theorem 3.9 n − 1 times, we can obtain that U ∗ AU is diagonal. For square matrices with dimension n ≥ 5, n − 2 or fewer eigenvalues may be on the boundary, and they don’t have to be normal. Later in this thesis, we’ll do experiments where we have a 5 × 5 square matrix which is almost normal, but for a small interference number r > 0 still has the numerical range as the convex set where some of the eigenvalues are vertices, and the rest of them interior points..

(54)

(55) Chapter 4. Gershgorin Discs This chapter contains useful theorems about Gershgorin discs. Understanding fundamental theory behind numerical range, one can interpret these regions. A general reference is Gustafson [2]. We start here by stating the definition of Gershgorin discs. n o 0 Definition 4.1 Let A ∈ Mn . Then the (closed) discs G (i) (A) = |z − aii | ≤ Ri (A) ⊂ P

(56)

(57) 0 C, ∀i = 2, . . . , n, where Ri (A) = j6=i

(58) aij

(59) interpreted as the row sum, are called Sn the Gershgorin discs, and G(A) = i=1 G (i) (A), the Gershgorin region. An example is in place, in order to understand the definition.   5 0 4 Example 4.2 Consider the matrix A = 1 3 3 . This matrix has the 0 1 5 √ √ 1 1 eigenvalues λ1 = 4, λ2 = 2 (9 + 17), λ3 = 2 (9 − 17). The Gershgorin discs are . . . G (1) (A) = |z − 5| ≤ 4 , G (2) (A) = |z − 3| ≤ 4 , G (3) (A) = |z − 5| ≤ 1 . Also, the Gershgorin region is the union of discs. Erik Jonsson, 2019.. 21.

(60) 22. Chapter 4. Gershgorin Discs. Figure 4.1: Gershgorin region of A in Example 4.2.

(61) 23. We see clearly here that G (3) (A) is inside the disc G (1) (A), they have the same centre, but different radius. Note that in the picture above, blue diamonds represents the centre of respective disc and red boxes the eigenvalues of A. Theorem 4.3 Let A ∈ Mn . Then the relation σ(A) ⊂ G(A) holds. [2] x = λ¯ x, x ¯ ∈ Cn , x ¯ 6= ¯0. Proof: Let λ be an eigenvalue

(62)

(63) of A, then λ ∈ σ(A),PA¯ n

(64)

(65) Choose k s.t. |xk | = max xj . Now we have λxk = i=1 aki xi , which we can 1≤j≤n P write as λxk − akk xk = i6=k aki xi . P P Therefore, |λ − akk ||xk | ≤ i6=k |aki ||xi | ≤ i6=k |aki ||xk |. Hence, |λ − akk | ≤ P (k) (A) ⊂ G(A). i6=k |aki | and λ ∈ G The proof can also be done by using theSsame argument with counting col0 0 0 n umn sums of A, thenowe aquire σ(A) ⊂ i=1 G (i) = G (A), where G (A) = n P

(66)

(67) |z − aii | ≤ j6=i

(68) aj i

(69) . The theorem above tells us that σ(A), the spectrum of A, lies somewhere in the Gershgorin region, G(A), the union of all Gershgorin discs. If the Gershgorin region is the disjoint union of sets, then the number of eigenvalues in each set equals the number of discs that form it. Corollary 4.4 Let A ∈ Mn . Then the relation σ(A) ⊂ G(A) ∩ G(AT ) holds. 0. Proof: Since σ(A) = σ(AT ) ⊂ G(AT ) = G (A), then from Theorem 4.3, it’s clear that σ(A) ⊂ G(A) ∩ G(AT ). We’ll do an example of illustrating G(A) ∩ G(AT ). 4 7 . This matrix has Example 4.5 Consider the real-valued matrix A = 2 6 √ √ the eigenvalues λ1 = 5 + 15, λ2 = 5 − 15. Illustrating the Gershgorin regions G(A) and G(AT ) below.

(70) 24. Chapter 4. Gershgorin Discs. Figure 4.2: The intersection of G(A) and G(AT ) in Example 4.5 where the region in blue is G(A) and the one in red is G(AT ). It’s clear that σ(A) ⊂ G(A) ∩ G(AT ) (observe that the black diamonds are the centres of the discs and blue boxes eigenvalues of A and AT )..

(71) Chapter 5. Results In this chapter, we investigate different types of complex-valued matrices by doing some illustrations in Maple, see appendix for main code. An observation before going through the theory below, is that the command from Maple code, W works well, but doesn’t draw enough points for the numerical range of a square matrix, which makes the boundary of the region non-smooth. To make sure that the boundary of the numerical becomes smooth, we need to apply a second argument in the following commandW(A,p), where p is the number of points to draw the boundary of the numerical range, and A is a square matrix. By setting p to a huge number (default setting is 65 which is not enough for a smooth boundary of the numerical range), the numerical range becomes more smooth. For drawing the numerical range of a square matrix in the experiments, p is set to 800. Another remark is that the number of lines that are being used for drawing the envelope for the numerical range is 120. Also, the number of points used for the envelope is 1000.. 5.1. Theory behind the illustrations. In this section, we motivate the illustrations of the numerical range of the matrices by discussing the theory behind the code which is based on results from Johnson [3]. Firstly, provided that we understand what an envelope is (check in PrelimErik Jonsson, 2019.. 25.

(72) 26. Chapter 5. Results. inaries chapter to be sure), let A=. A + A∗ A − A∗ + , 2 2. ∗. where H(A) = A+A is the Hermitian part of A. We realize that H(A) is 2 self-adjoint, so from before, we know that H(A) has real eigenvalues. Now consider µ1 as the largest eigenvalue of H(A) and λ an eigenvalue of A, with normalized eigenvector x ¯. Then Re(λ) =. 1 ¯ = 1 (¯ (λ + λ) x∗ A¯ x+x ¯ ∗ A∗ x ¯) = x ¯∗ H(A)¯ x ≤ µ1 . 2 2. This means the spectrum of A stretches to the left of the vertical line that goes through µ1 (analogously if we denote the smallest eigenvalue of H(A), µ2 , then the spectrum of A stretches to the right of the vertical line that goes through µ2 ). Now, by denoting Aθ = eiθ A, we understand that Aθ must have eigenvalues λj eiθ if A has eigenvalues λj . Also, A +A∗ denote H(Aθ ) = θ 2 θ with largest eigenvalue λθ = µ1 (Aθ ). So, by refering to −iθ a result in [3], denote the half-planes {z : Re −iθ Hθ ≡ e z ≤ λθ }, ∀θ ∈ [0, 2π), generated by straight lines Lθ ≡ e (λθ + ti), ∀t ∈ R , where λθ is the largest eigenvalue of Hθ . By convexity of F(A), this leads us to a result in [3, Section 1.5] F(A) =. \. 1. Hθ .. θ∈[0,2π). The numerical range is the intersection of all Hθ , which means the boundary of F(A) is the envelope of the family of lines Lθ , ∀θ ∈ [0, 2π). Also, the complex number aθ ≡ x ¯∗θ A¯ xθ is a boundary point of F(A), given the prerequisites above, where x ¯θ is a normalized eigenvector to H(Aθ ) and λθ .. 1A. result which we motivate later in the upcoming subsection..

(73) 5.2. General matrices in M2. 5.1.1. 27. Drawing the illustrations. The theory above, gives the reader an idea how to illustrate the numerical range of general square matrices in any calculation program, i.e. Maple, Matlab, et cetera. The approach is to draw lines Lθ , ∀θ ∈ [0, 2π), so the numerical range of any square matrix is obtained from the intersection of all Hθ (the boundary of the numerical range is the envelope of a family of lines Lθ ). Note that in the upcoming experiments, the axes are denoted s and t in the figures.. 5.2. General matrices in M2. 2 Consider the matrix A1 = 0. 3 . It has the eigenvalues λ1 = 2, λ2 = 1. 1. The Maple code gives us the following numerical range. Figure 5.1: Numerical range of A1. Figure 5.2: Envelope for F(A1 ).

(74) 28. Chapter 5. Results. which is definitely the region in the complex plane bounded by an ellipse. The ellipse has the centre ( 32 , 0), minor semi-axis ± 23 , major semi-axis q √ √ 2 ± 34 + ( 12 )2 = ± 210 and vertices at ( 3±2 10 , 0) on the s-axis. Also, λ1 and λ2 are interior points in the ellipse. Note that A1 already is on Schur form. 1 1 We look at another matrix, A2 = which has the eigenvalues λ1 = 2 3 √ √ 2 + 3, λ2 = 2 − 3. # " √ −1√ 2+ 3 The Schur form of A2 is Γ = , so we get instead the numerical 0 2− 3 range. Figure 5.3: Numerical range of A2. Figure 5.4: Envelope for F(A2 ).

(75) 5.2. General matrices in M2. 29. We see that the numerical range is similar to A1 , the region in the complex plane bounded by an ellipse. This ellipse has the centre (2, 0), minor semi-axis q √ 1 1 ± 2 , major semi-axis ± 4 + ( 3)2 ≈ 1.8 and vertices (2 ± 1.8, 0). Here, the eigenvalues of A2 are interior points, but quite close to the vertices. 5 1 Let’s look at the third example on a matrix A3 = . The matrix has the 0 −5 eigenvalues λ1 = 5, λ2 = −5. The numerical range is. Figure 5.5: Numerical range of A3. Figure 5.6: Envelope for F(A3 ).

(76) 30. Chapter 5. Results. A3 has as A1 and A2 the numerical range as the region in the complex plane bounded by an ellipse. Note thatqthis ellipse has the origin as centre, minor √. semi-axis ± 12 , major semi-axis ± 14 + 52 = ± 101 and vertices (± 2 eigenvalues are again close to the vertices, inside the ellipse).. 5.3. √. 101 2 , 0). (the. Normal matrices. In this section, we do experiments on normal matrices. We’ll examine the convex hulls and see how exactly they differ as regions in the complex plane. Also, illustrations of their envelopes are important in this matter. 2 5 Let’s start here with a normal (Hermitian) 2 × 2 matrix A4 = . The 5 1 matrix has the eigenvalues λ1 =. 3 2. √. +. 101 2 , λ2. =. 3 2. √. −. 101 2 .. The numerical range is the convex hull, which is the line segment. Figure 5.7: Numerical range of A4. Figure 5.8: Envelope for F(A4 ).

(77) 5.3. Normal matrices. 31. We see here that the numerical range is the line segment joining the smallest and largest eigenvalue (according to Theorem 3.6, r = 0, a flat ellipse). Next case, we’ll look at a normal (Hermitian matrix  again) complex-valued  1 i 0 of dimension 3. Consider the matrix A5 = −i 1 0 . The matrix has the 0 0 1 eigenvalues λ1 = 0, λ2 = 1, λ3 = 2. The numerical range is the convex hull, which is the line segment. Figure 5.9: Numerical range of A5. Figure 5.10: Envelope for F(A5 ).

(78) 32. Chapter 5. Results. As we can see here, Hermitian matrices have a very odd behavior when it comes to their numerical range, i.e. the region in the complex plane bounded by a line segment joining the smallest and largest eigenvalue. We can also say that the convex hulls either are the line segment containing all of the real values forming the spectrum of the Hermitian matrices, or for more general normal matrices the region in the complex plane bounded by a polygon (triangle, quadriteral, et cetera).   1 0 0 0 0 0 1 0  Consider the normal matrix A6 =  0 0 0 1 . The matrix has the eigenval0 1 0 0 ues λ1 = − 12 +. √ i 3 2 , λ2. = − 12 −. √ i 3 2 , λ3. = 1, λ4 = 1.. The numerical range is the convex hull, which is the triangle. Figure 5.11: Numerical range of A6. Figure 5.12: Envelope for F(A6 ).

(79) 5.4. Almost normal matrices. 33. The numerical range is, indeed, the convex hull, the region in the complex plane bounded by a triangle. According to Theorem 3.10, the matrix must be normal (obviously since it’s stated before the illustrated pictures above), because its dimension is 4.. 5.4. Almost normal matrices. In this section, we study matrices and also other types of matrices.  4 0 Consider the matrix A7 = 0 2 0 0 1, λ2 = 2, λ3 = 4.. that are complex-valued and almost normal  0 3 . The matrix has the eigenvalues λ1 = 1. The numerical range is. Figure 5.13: Numerical range of A7. Figure 5.14: Envelope for F(A7 ).

(80) 34. Chapter 5. Results. By seeing the form of A7 , we realize that λ3 is on the boundary of the numerical range, and the rest of the eigenvalues interior points. Then, by L 2 3 Theorem 3.9, we have U ∗ A7 U = A7 = 4 , where U = I, unitary. 0 1 Therefore, the numerical range of A7 is the convex set of the eigenvalues since it’s a set of an oval and a normal part.. 5.4.1. Interference of interior points.   0 0 0 0 0   0 0 0  0 2 + 3i   0 6 − 5i 0 0  . This maNow, consider next matrix A8 = 0   0 0 2+i 0  0 5 0 0 0 0 2 −i trix is obviously normal and has the eigenvalues λ1 = 0, λ2 = 2 + 3i, λ3 = 6 − 5i, λ4 = 2 + i, λ5 = 52 − i. The numerical range is. Figure 5.15: Numerical range of A8. Figure 5.16: Envelope for F(A8 ).

(81) 5.4. Almost normal matrices. 35. According to Theorem 3.10, the numerical range of this matrix is the convex hull of the eigenvalues λ1 , λ2 and λ3 , the region in the complex plane bounded by a triangle with interior points λ4 and λ5 . But, the convex hull is also obtainedfor a small number r > 0 if the matrix looks like  0 0 0 0 0   0 0 0  0 2 + 3i   e8 (r) = 0 0 6 − 5i 0 0 . A   0 0 2+i r  0 5 0 0 0 0 2 −i However, for a large number enough, the numerical range would look different. Take r = 6, then the envelope now is. e8 ) when r = 6 Figure 5.17: Envelope for F(A.

(82) 36. Chapter 5. Results. The larger r is, the more oval the boundary of the numerical range become to be. This matrix is no longer normal, though. It’s also interesting to see here that the numerical ranges of both matrices are convex sets (obvious now to mention that the oval part of the numerical range e8 is flat for a small number r > 0). of A. 5.4.2. Interference of boundary points. It’s interesting to see how sensitive A8 is, if we investigate the interference around a boundary point this time.   0 r 0 0 0   0 0 0  0 2 + 3i   0 6 − 5i 0 0 , where Firstly, we define the matrix Aˆ8 (r) = 0   0 0 2+i 0  0 5 0 0 0 0 2 −i this matrix actually is A8 but applied with a interference number r around a boundary point this time. For a small r, r = 38 , the envelope now is. Figure 5.18: Envelope for F(Aˆ8 ) when r =. 3 8.

(83) 5.4. Almost normal matrices. 37. One of the edges for the boundary of the numerical range is a bit oval, so it appears for a small number, the interference isn’t large enough. However, looking at the point λ2 = 2 + 3i, it’s not clear if it’s still a vertex on the region in the complex plane bounded by the triangle. By investigating this further, see the envelope of the same triangle but the coordinate axes are around λ2 below. Figure 5.19: Envelope for F(Aˆ8 ), the point λ2 = 2 + 3i is zoomed.

(84) 38. Chapter 5. Results. It’s now easier to see that λ2 is still a vertex of the triangle. Let’s see what happens if we raise the number further, actually greater than 1.   0 5 0 0 0   0 0 0  0 2 + 3i   0 6 − 5i 0 0 , the envelope is For Aˆ8 (5) = 0   0 0 2+i 0  0 5 0 0 0 0 2 −i. Figure 5.20: Envelope for F(Aˆ8 ) when r = 5.

(85) 5.5. Other matrices that are not normal. 39. The region is, at this point, very oval like the numerical range of A7 . It’s clear from now on that the numerical range becomes more oval if we raise the interference number r any further.. 5.5. Other matrices that are not normal.   0 1 0 Next, we look at a matrix A9 = 0 0 1 , which appears to be a nilpotent 0 0 0 matrix (a matrix A is nilpotent if AN = 0, N ≥ 1). Then the eigenvalues are λ1 = λ2 = λ3 = 0. The numerical range is. Figure 5.21: Numerical range of A9. Figure 5.22: Envelope for F(A9 ).

(86) 40. Chapter 5. Results. It appears that the nilpotent matrix has its numerical range as the region in the complex plane bounded by a circle with the eigenvalues asinterior points.  0 a 0 However, looking at the matrix again, but Aˆ9 = 0 0 b  , a, b ∈ C. a or b 0 0 0 can be zero, but not both (otherwise we’ll have a normal matrix and the numerical range result to be the point where the eigenvalues join). Raising any of a or b would by earlier experiments on other matrices result to the region bounded by a circle with greater radius, around origin. To be convinced, let’s raise a.   0 8 0 e9 = 0 0 1. The numerical range is now Now, the matrix is A 0 0 0. e9 Figure 5.23: Numerical range of A. e9 ) Figure 5.24: Envelope for F(A.

(87) 5.6. Gershgorin region versus Numerical range. 5.6. 41. Gershgorin region versus Numerical range. In this section, we study the Gershgorin regions and numerical ranges of several square matrices, and compare which of them is better (or perhaps both good) in bounding the spectrum.. 5.6.1. Matrices in M2 . 1 First, consider the matrix A10 . This matrix has the following eigen0 −2 3 values λ1 = −2, λ2 = 2 and the matrix Γ1 = is the Schur form of A10 . 0 2 0 = 4. The Gershgorin region and the envelope for the numerical range are. Figure 5.25: Envelope for F(A10 ). Figure 5.26: Gershgorin region for A10.

(88) 42. Chapter 5. Results. A comment to the numerical range, it’s the region in the complex plane bounded by an ellipse with origin as centre, minor semi-axis ± 32 and major q 2 semi-axis ± 34 + (−2)2 = ± 52 . We see here that the Gershgorin region is the region in the plane bounded by circles where the smaller circle is inside the bigger circle. If we look at the following intersection of these regions below,. Figure 5.27: Intersection of F(A10 ) and G(A10 ).

(89) 5.6. Gershgorin region versus Numerical range. 43. we can see here that the numerical range is completely inside the Gershgorin region (small circle inside the ellipse though). We see here that σ(A10 ) lies inside both regions in the plane. Therefore, we can conclude that the numerical range is the best set bounding the spectrum here. " # −1 21 Let’s look at next matrix. Consider A11 = 1 . This matrix has the 1 4 # " √ √. following eigenvalues λ1 = − 3 4 2 , λ2 =. √ 3 2 4. and the matrix Γ2 =. 3 2 4. 0. is the Schur form of A11 . The Gershgorin region and the envelope for the numerical range are. Figure 5.28: Envelope for F(A11 ). Figure 5.29: Gershgorin region for A11. 1 4√ −3 2 4.

(90) 44. Chapter 5. Results. A comment to the numerical range, it’s the region in the complex plane bounded by an ellipse with origin as centre, minor semi-axis ± 18 and major q 1 √ 3 2 2 + ( semi-axis ± 16 4 4 ) ≈ ±1.068. We see here that the Gershgorin region is the region in the plane bounded by circles where the circles neither intersect or are inside of another. If we look at the following intersection of these regions below,. Figure 5.30: Intersection of F(A11 ) and G(A11 ).

(91) 5.6. Gershgorin region versus Numerical range. 45. we can see here that parts of the numerical range are inside the Gershgorin region. This means that the rest of the numerical range is cut off by the Gershgorin region. We see here that σ(A11 ) lies inside both regions in the plane. Therefore, we can conclude that the intersection of these regions is the best set for bounding the spectrum. The last experiment we’ll examine regarding the dimension 2 case, is the follow−1 1 ing normal matrix A12 = . This matrix has the following eigenvalues 1 1 # "√ √ √ 2 0 √ is the Schur form of A12 . λ1 = − 2, λ2 = 2 and the matrix Γ3 = 0 − 2 The Gershgorin region and the envelope for the numerical range are. Figure 5.31: Envelope for F(A12 ). Figure 5.32: Gershgorin region for A12.

(92) 46. Chapter 5. Results. A comment to the numerical range, it’s a line segment joining the smallest and largest eigenvalue (a flat ellipse for r = 0). We see here that the Gershgorin region is the region in the complex plane bounded by circles where the circles intersect in origin. If we look at the following intersection of these regions below,. Figure 5.33: Intersection of F(A12 ) and G(A12 ).

(93) 5.6. Gershgorin region versus Numerical range. 47. we see that the whole line segment lies in the union of the circles where σ(A12 ) lies within. Therefore, we can conclude that the numerical range is better here.. 5.6.2. Matrices in M3 and higher dimensions. In this subsection, we study matrices in higher dimensions in the matter of comparing numerical ranges to Gershgorin regions.   −4 0 1 Consider the matrix A13 =  0 1 4. This matrix has the eigenvalues −1 0 6 √ √ λ1 = 1, λ2 = 1 − 2 6, λ3 = 1 + 2 6. The Gershgorin region and the envelope for the numerical range are. Figure 5.34: Envelope for F(A13 ). Figure 5.35: Gershgorin region for A13.

(94) 48. Chapter 5. Results. The numerical range is an oval region in the complex plane. The Gershgorin region is a kind of the situation for matrix A12 , where the bigger circle intersect the smaller circles. If we look at the following intersection of these regions below,. Figure 5.36: Intersection of F(A13 ) and G(A13 ).

(95) 5.6. Gershgorin region versus Numerical range. 49. we see that parts of the oval region and the Gershgorin region intersects. Also, the Gershgorin region cuts off the rest of the oval region. σ(A13 ) lies in the intersection of the regions in the plane. Therefore, we can conclude that the intersection of these regions is the best set for bounding the spectrum.   6+i 0 1 3 2 . This matrix has the Let’s look at the next matrix A14 =  0 3 0 −2 + i √ √ eigenvalues λ1 = 3, λ2 = 2 + 19 + i, λ3 = 2 − 19 + i. The Gershgorin region and the envelope for the numerical range are. Figure 5.37: Envelope for F(A14 ). Figure 5.38: Gershgorin region for A14.

(96) 50. Chapter 5. Results. Like the previous matrix, the numerical range looks like an oval region in the complex plane. However, the Gershgorin region is a kind of the situation for matrix A11 , circles not intersecting anywhere in the plane. If we look at the following intersection of these regions below,. Figure 5.39: Intersection of F(A14 ) and G(A14 ).

(97) 5.6. Gershgorin region versus Numerical range. 51. we see that parts of the oval region and the Gershgorin region intersects. Also, the Gershgorin region cuts off the rest of the oval region. σ(A14 ) lies in the intersection of the regions in the plane. Therefore, we can conclude that the intersection of the regions is the best set for bounding the spectrum.   1 0 1   1 . One last experiment for the 3×3 case, consider the matrix A15 = 0 1+i 4 1 1 0 −2 It has the eigenvalues λ1 = 32 , λ2 = 1+i 4 , λ3 = −1. The Gershgorin region and the envelope for the numerical range are. Figure 5.40: Envelope for F(A15 ). Figure 5.41: Gershgorin region for A15.

(98) 52. Chapter 5. Results. Like the previous matrix, the numerical range looks like an oval region in the complex plane. However, the Gershgorin region is the union of the three circles with radius 1 but different centres. If we look at the following intersection of these regions below,. Figure 5.42: Intersection of F(A15 ) and G(A15 ).

(99) 5.6. Gershgorin region versus Numerical range. 53. we see that σ(A15 ) lies in both regions in the plane. However, we can conclude that the oval region lies inside the union of the circles, so the numerical range is the best set here. We’ll now look  0 − 74   1 − 14 + i  0  0  0  1 0 0. at matrices ofdimension n > 3. Consider the matrix A16 = 0 1 0  0 0 0  5 1 0  . This matrix has the eigenvalues 4  11 0 4 +i 0 17 1 0 4. λ1 = − 14 + i, λ2 =. 1+i 2. √. −. 93+36i , λ3 4. =. 1+i 2. √. +. 93+36i , λ4 4. = 54 , λ5 =. 17 4 .. The Gershgorin region and the envelope for the numerical range are. Figure 5.43: Envelope for F(A16 ). Figure 5.44: Gershgorin region for A16.

(100) 54. Chapter 5. Results. Like the previous matrix, the numerical range looks like an oval region in the complex plane. However, the Gershgorin region is the union of five circles with the same radius 1 but has different centres (it looks kind of the symbol of Olympic Games). If we look at the following intersection of these regions below,. Figure 5.45: Intersection of F(A16 ) and G(A16 ).

(101) 5.6. Gershgorin region versus Numerical range. 55. we see that parts of the oval region and the Gershgorin region intersects. σ(A16 ) lies in the intersection of the regions in the plane. Therefore, we can conclude that the intersection of the numerical range and Gershgorin region is the best suited set for bounding the spectrum.   −2 1 0 0 0 0 0 1 0 0 0 0 0 0    0 1 2 0 0 0 0    . 0 0 Consider the matrix A17 =   0 0 1 −2 + 2i  0 0 0  1 −2 − 2i 0 0   0 0 0 0 1 2 + 2i 0  0 0 0 0 0 1 2 − 2i This matrix has the eigenvalues. λ1 = −2 + 2i, λ2 = −2 − 2i, λ3 = −1 − 2 + 2i, λ7 = 2 − 2i.. √. 2, λ4 = −1 +. √. 2, λ5 = 2, λ6 =. The Gershgorin region and the envelope for the numerical range are.

(102) 56. Chapter 5. Results. Figure 5.46: Envelope for F(A17 ). Figure 5.47: Gershgorin region for A17.

(103) 5.6. Gershgorin region versus Numerical range. 57. The numerical range is an oval region in the complex plane (looks kind of a pillow). Also, the Gershgorin region is the union of seven circles with the same radius 1 but has different centres (the union forms a H). If we look at the following intersection of these regions below,. Figure 5.48: Intersection of F(A17 ) and G(A17 ).

(104) 58. Chapter 5. Results. we see that parts of the oval region and the Gershgorin region intersects. It’s easy to see that the Gershgorin region cuts off the rest of the oval region. σ(A17 ) lies in the intersection of the regions in the plane. Therefore, we can conclude that the intersection of the numerical range and Gershgorin region is the best suited set for bounding the spectrum.   0 1 0 0 0 0 0 0 0 0 −2 1 0 0 0 0 0 0    0 0 −4 1 0 0 0  0 0   0 0  0 2 1 0 0 0 0     . This ma0 0 0 0 4 1 0 0 0 Consider the matrix A18 =   0 0  0 0 0 2i 1 0 0   0 0  0 0 0 0 4i 1 0   0 0 0 0 0 0 0 −2i 1  0 0 0 0 0 1 0 0 −4i trix has the eigenvalues p p √ √ λ1 = −4, λ2 =p−2, λ3 = −i 10p+ 37, λ4 = −i 10 − 37, λ5 = 0, λ6 = √ √ 2, λ7 = 4, λ8 = i 10 − 37, λ9 = 10 + 37. The Gershgorin region and the envelope for the numerical range are.

(105) 5.6. Gershgorin region versus Numerical range. Figure 5.49: Envelope for F(A18 ). Figure 5.50: Gershgorin region for A18. 59.

(106) 60. Chapter 5. Results. The numerical range is an oval region in the complex plane (looks like a non-smooth rhombus). Also, the Gershgorin region is the union of nine circles with the same radius 1 but has different centres (the union forms a plus). If we look at the following intersection of these regions below,. Figure 5.51: Intersection of F(A18 ) and G(A18 ). we see that parts of the oval region and the Gershgorin region intersects. It’s also easy to see that the Gershgorin region cuts off the rest of the numerical range. σ(A18 ) lies in the intersection of the regions in the plane. Therefore, we can conclude that the intersection of the numerical range and Gershgorin region is the best suited set for bounding the spectrum..

(107) Chapter 6. Discussion In this chapter, we discuss the results and give a conclusion about numerical range and Gershgorin region for a square matrix, and also mention some topics that remain to investigate.. 6.1. Numerical range. In the result chapter, we have illustrated different cases of numerical ranges of square matrices, whether they be normal or not. The most important results are three. Firstly, is that the numerical range of a normal square matrix is either for Hermitian matrices, the line segment joining the smallest and largest eigenvalue and the rest of the eigenvalues are interior points, or more general normal matrices the region in the complex plane bounded by a polygon where the eigenvalues are vertices. Secondly, for the non-normal matrices, we have stated before, that their numerical range is an oval region in the plane (or in 2 × 2 case the region bounded by an ellipse). Finally, in the special case when the numerical range of an arbitrary square matrix is a combination of a normal and oval part, whether if there are 1 or 2 eigenvalues as interior points and the rest of the eigenvalues are vertices, we have seen that the region in the complex plane becomes more oval by raising the interference number r. However, for small numbers, the region become closely Erik Jonsson, 2019.. 61.

(108) 62. Chapter 6. Discussion. the convex set (2 or more eigenvalues might be interior points if the dimension is greater than 4). If the dimension of the square matrix n ≤ 4, then the numerical range become closely the convex hull.. 6.2. Gershgorin regions. In Chapter 4, we examined analytically Gershgorin regions of general square matrices. In the result section, we have seen different types of Gershgorin regions containing the spectrum of a square matrix if the Gershgorin region is obtained by determining the union of Gershgorin discs (where every Gershgorin disc is aquired from each row in the matrix). However, the Gershgorin region can be obtained by the union of Gershgorin discs where every disc is determined from each column in the matrix, which means the spectrum of this matrix lies in the intersection of the Gershgorin regions. Finally, it’s indeed true that the number of eigenvalues in each Gershgorin disc equals the number of the Gershgorin discs that forms the Gershgorin region, the disjoint union of discs.. 6.3. Numerical range versus Gershgorin regions. As it has seen in some of the later experiments, the numerical range has shown to be the best set for bounding the spectrum. In other cases, however, the Gershgorin region has been useful in the sense of cutting off the parts of numerical range which doesn’t overlap within the Gershgorin region. This results to that the spectrum lies in the intersection of the numerical range and the Gershgorin region, so in a sense both sets are great for bounding the spectrum of the matrix.. 6.4. Further topics to investigate. We’ve covered the most fundamental theorems in the theory behind numerical range and also Gershgorin region. However, there is much more to discover, i.e. theorems that describes other types of regions containing the spectrum of any square matrix, like Ostrowski and Brauer regions (see [14], [10] for a brief summary of their definitions), which are based on the theory behind the Gershgorin disc theorem. In a future research, the interest might lie in how to interpret the Ostrowski and Brauer regions with the help of understanding Gershgorin discs. This can be examined by the usage of functional analysis, defining the fundamental theory behind numerical range and Gershgorin region more generally by using different types of operators, but the subject itself is also achievable by examining matrices.

(109) 6.4. Further topics to investigate. 63. that maps Cn → Cn . However, it’s a whole new world to explore applications in spectral theory, especially in analysis of spectra of general matrices (or operators defined in other ways)..

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(111) Bibliography [1] Böiers, Lars-Christer and Persson, Arne. Analys i flera variabler. 3:10 Edition. Studentlitteratur AB, Lund, 2005. [2] Gustafson, Karl.E and Rao, Duggirala K.M. Numerical range. SpringerVerlag New York, Inc. 1997. [3] Horn, Roger A. and Johnson, Charles R. Topics in matrix analysis. Cambridge University Press, USA, 1991. [4] Janfalk, Ulf. Linjär algebra. Matematiska institutionen, Linköpings universitet, Linköping, 2012. [5] Lundgren, Jan; Rönnqvist, Mikael and Värbrand, Peter. Optimization. 1:2 Edition. Studentlitteratur AB, Lund, 2012. [6] Maplesoft. Gershgorin Circle Theorem. 2016. https://www.maplesoft. com/applications/view.aspx?sid=4814&view=html (Retrieved 2019-0401) [7] Maplesoft. Numerical range of a square matrix. 2019. https: //www.maplesoft.com/applications/view.aspx?sid=4128&view=html (Retrieved 2019-03-25) [8] Shapiro, Helene. Linear algebra and matrices. American Mathematic Society, Providence, USA, 2015. [9] Treil, Sergei. Linear algebra done wrong. Department of Mathematics, Brown University, 2017. http://www.math.brown.edu/~treil/papers/ LADW/LADW_2017-09-04.pdf [10] Wolfram MathWorld. Brauer’s Theorem. 2019. http://mathworld. wolfram.com/BrauersTheorem.html (Retrieved 2019-04-10) Erik Jonsson, 2019.. 65.

(112) 66. Bibliography. [11] Wolfram MathWorld. Envelope. 2019. http://mathworld.wolfram.com/ Envelope.html (Retrieved 2019-03-21) [12] Wolfram MathWorld. Frobenius Norm. 2019. http://mathworld.wolfram. com/FrobeniusNorm.html (Retrieved 2019-03-25) [13] Wolfram MathWorld. Matrix Direct Sum. 2019. http://mathworld. wolfram.com/MatrixDirectSum.html (Retrieved 2019-05-28) [14] Wolfram MathWorld. Ostrowski’s Theorem. 2019. http://mathworld. wolfram.com/OstrowskisTheorem.html (Retrieved 2019-04-10).

(113) Appendix A. Maple Commands(Code) You’ll find valuable Maple commands in case that something is unclear with the simulations. The code for the command W in section A.1, however, isn’t attached here, so I recommend you check the hyperlink for the source code in the bibliography [7]. One observation is that W is based on obtaining the envelope for the numerical range of general square matrices, but leaving the lines out of it. In section A.2, completed Maple code has been used for generating the Gershgorin regions. However, the code that generates the intersection of G(A) and G(A∗ ) is modified from the complete Maple code. Commands and the modified code are attached in section A.2 (see [6] for source code of Gershgorin regions).. A.1. Numerical range. with(plots): restart:. with(LinearAlgebra):. . A := 2,0 | 3,1 : Eigenvalues(A): F:= Array(1..121): for j from 1 by 1 to 120 do A := evalf(exp((2.0.*Pi*I)/120)*A); H :=(A + A∗ )/2 : eig := Eigenvectors(H, output = list): eig := sort(eig, (a,b) ->is(Re(a[1]) > Im(b[1]))): Erik Jonsson, 2019.. 67.

(114) 68. Appendix A. Maple Commands(Code). d1:= eig[1,1]: F[j] = implicitplot(d1 - cos((j*2*Pi)/120)*s +sin((j*2*Pi)/120)*t = 0, s = -4..4, t = -3..3,numpoints = 1000): end do: ev := Eigenvalues(A): F[121] := pointplot([[Re(ev[1]), Im(ev[1])], [Re(ev[2]), Im(ev[2])]], symbol = box, color = "Blue"): display(seq(F[j], j = 1..121)): W(A,800):. A.2. Gershgorin region. Gershgorin(A): Gershgorin(A∗ ) : GershgorinIntersect := proc (A::Matrix, B::Matrix) local Delta, Lambda, m, n, AA, BB, R, Rho, C, C_star, i, c, d, eig1, eig2, P, Q, Plt1, Plt2; Delta := proc (i, j) if i = j then 0 else 1 end if end proc; Lambda := proc (i, k) if i = k then 0 else 1 end if end proc; m, n := LinearAlgebra[Dimension](A); m, n := LinearAlgebra[Dimension](B); AA := Matrix(m,n,(i,j) → Delta(i,j)abs∗ (A[i,j])): BB := Matrix(m,n,(i,k) → Delta(i,k)abs∗ (B[i,k])): R := evalm(&∗ (AA, Vector(m, 1))): ∗ Rho := evalm(& (BB, Vector(m, 1))): C := seq(’plottools[circle]’([Re(A[i,i]),Im(A[i,i])],. R[i],color=blue,i=1..m) : C_star := seq(’plottools[circle]’([Re(B[i,i]),Im(B[i,i])], Rho[i],. color=blue,i=1..m) ; c := seq(’plottools[point]’([Re(A[i,i]),Im(A[i,i])], color=black,. symbol=diamond),i=1..m) ; d := seq(’plottools[point]’([Re(B[i,i]),Im(B[i,i])], color=green,. symbol=diamond),i=1..m) ; eig1 := evalf(LinearAlgebra[Eigenvalues](A)); eig2 := evalf(LinearAlgebra[Eigenvalues](B)); P := seq(’plottools[point]’([Re(eig1[i]), Im(eig1[i])], color=red,. symbol=box),i=1..m) ;.

(115) A.2. Gershgorin region. 69. Q := seq(’plottools[point]’([Re(eig2[i]), Im(eig2[i])], color=blue,. symbol=box),i=1..m) ; Plt1 := C union c union P; Plt2 := C_star union d union Q; plots[display](eval(Plt1),eval(Plt2), scaling=constrained);end: GershgorinIntersection(A,A∗ ): display(W(A,800),Gershgorin(A)):.

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