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Numerical Range of Square Matrices a Study in Spectral Theory 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: G2 Supervisor: Göran Bergqvist, Department of Mathematics, Linköping University Examiner: Milagros Izquierdo, Department of Mathematics, Linköping University Linköping: June 2019 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 Sammanfattning I denna uppsats diskuterar vi viktiga resultat för numerical range av gene- rella 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 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 inter- esting 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 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 Contents List of Figures xiii 1 Introduction 1 2 Preliminaries 3 2.1 Spectral theory . 3 2.2 Fundamental concepts from Optimization and Multi-variable Cal- culus . 7 3 Numerical Range 9 3.1 Numerical range of general square matrices . 9 3.2 Numerical range of a 2 × 2 matrix . 12 3.3 Topology behind the numerical range of a general matrix . 17 3.4 General results of normal matrices . 18 4 Gershgorin Discs 21 5 Results 25 5.1 Theory behind the illustrations . 25 5.1.1 Drawing the illustrations . 27 5.2 General matrices in M2 ....................... 27 5.3 Normal matrices . 30 5.4 Almost normal matrices . 33 5.4.1 Interference of interior points . 34 5.4.2 Interference of boundary points . 36 5.5 Other matrices that are not normal . 39 5.6 Gershgorin region versus Numerical range . 41 5.6.1 Matrices in M2 ........................ 41 5.6.2 Matrices in M3 and higher dimensions . 47 Erik Jonsson, 2019. xi xii Contents 6 Discussion 61 6.1 Numerical range . 61 6.2 Gershgorin regions . 62 6.3 Numerical range versus Gershgorin regions . 62 6.4 Further topics to investigate . 62 Bibliography 65 A Maple Commands(Code) 67 A.1 Numerical range . 67 A.2 Gershgorin region . 68 List of Figures 3.1 Numerical range of A in Example 3.5 . 12 3.2 Numerical range of A in Example 3.7 . 16 4.1 Gershgorin region of A in Example 4.2 . 22 4.2 The intersection of G(A) and G(AT ) in Example 4.5 . 24 5.1 Numerical range of A1 ...................... 27 5.2 Envelope for F(A1) ........................ 27 5.3 Numerical range of A2 ...................... 28 5.4 Envelope for F(A2) ........................ 28 5.5 Numerical range of A3 ...................... 29 5.6 Envelope for F(A3) ........................ 29 5.7 Numerical range of A4 ...................... 30 5.8 Envelope for F(A4) ........................ 30 5.9 Numerical range of A5 ...................... 31 5.10 Envelope for F(A5) ........................ 31 5.11 Numerical range of A6 ...................... 32 5.12 Envelope for F(A6) ........................ 32 5.13 Numerical range of A7 ...................... 33 5.14 Envelope for F(A7) ........................ 33 5.15 Numerical range of A8 ...................... 34 5.16 Envelope for F(A8) ........................ 34 5.17 Envelope for F(Ae8) when r = 6 . 35 ˆ 3 5.18 Envelope for F(A8) when r = 8 . 36 5.19 Envelope for F(Aˆ8), the point λ2 = 2 + 3i is zoomed . 37 5.20 Envelope for F(Aˆ8) when r = 5 . 38 5.21 Numerical range of A9 ...................... 39 5.22 Envelope for F(A9) ........................ 39 5.23 Numerical range of Ae9 ...................... 40 Erik Jonsson, 2019. xiii xiv List of Figures 5.24 Envelope for F(Ae9) ........................ 40 5.25 Envelope for F(A10) ........................ 41 5.26 Gershgorin region for A10 .................... 41 5.27 Intersection of F(A10) and G(A10) . 42 5.28 Envelope for F(A11) ........................ 43 5.29 Gershgorin region for A11 .................... 43 5.30 Intersection of F(A11) and G(A11) . 44 5.31 Envelope for F(A12) ........................ 45 5.32 Gershgorin region for A12 .................... 45 5.33 Intersection of F(A12) and G(A12) . 46 5.34 Envelope for F(A13) ........................ 47 5.35 Gershgorin region for A13 .................... 47 5.36 Intersection of F(A13) and G(A13) . 48 5.37 Envelope for F(A14) ........................ 49 5.38 Gershgorin region for A14 .................... 49 5.39 Intersection of F(A14) and G(A14) . 50 5.40 Envelope for F(A15) ........................ 51 5.41 Gershgorin region for A15 .................... 51 5.42 Intersection of F(A15) and G(A15) . 52 5.43 Envelope for F(A16) ........................ 53 5.44 Gershgorin region for A16 .................... 53 5.45 Intersection of F(A16) and G(A16) . 54 5.46 Envelope for F(A17) ........................ 56 5.47 Gershgorin region for A17 .................... 56 5.48 Intersection of F(A17) and G(A17) . 57 5.49 Envelope for F(A18) ........................ 59 5.50 Gershgorin region for A18 .................... 59 5.51 Intersection of F(A18) and G(A18) . 60 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 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) = λu.¯ 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 Au¯ = λu¯. 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 4 Chapter 2. Preliminaries 4 9 Example 2.2 Consider the following matrix A = . Calculations by 9 4 using the definition of secular polynomial for A gives us 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 Au¯1 = λ1u¯1,Au¯2 = λ2u¯2. If it exist a basis of eigenvectors [u¯1, u¯2,..., u¯n] that corresponds to the eigenval- −1 ues of A, λ1, λ2, ..., λn, then A = T DT , where T has u¯1, u¯2,..., u¯n as columns.
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