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Matrix Theory

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378 kr

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Exkl. moms: 357 kr

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A good understanding of matrices and their properties is a necessary prerequisite for progress in almost any field within pure or applied mathematics, for example calculus in several variables, numerical analysis or control theory. This book is based on the course Matrix theory given at Lund University. It starts by recalling the basic theory of matrices and determinants, and then proceeds to more advanced subjects such as the Jordan Normal Form, functions of matrices, norms, normal matrices ...

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A good understanding of matrices and their properties is a necessary prerequisite for progress in almost any field within pure or applied mathematics, for example calculus in several variables, numerical analysis or control theory. This book is based on the course Matrix theory given at Lund University. It starts by recalling the basic theory of matrices and determinants, and then proceeds to more advanced subjects such as the Jordan Normal Form, functions of matrices, norms, normal matrices and singular values. The book may be used for a second course in linear algebra in a bachelor’s program in mathematics, or for a first year graduate course in engineering subjects. Distinguishing qualities compared with other texts on the subject are • The book starts gently compared with other texts with the same scope. • There are many carefully worked out examples. • It is possible to use the book for classes on different levels, by selecting parts of the material. The book contains a large number of exercises with answers, indication of difficulty and sometimes hints. The exercises are intended to help the students to think in new ways and to understand the art of proving mathematical statements.

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      • 9
        CHAPTER 1 Matrices – Conventions and Notations
        • 1.1
          9
          Matrices: long and short notations
        • 1.2
          11
          Block matrices
        • 1.3
          15
          Matrices as numbers Addition and subtraction
        • 1.4
          16
          Multiplication by numbers Some special classes of matrices
        • 1.5
          18
          Matrix multiplication
        • 1.6
          22
          Multiplication by elementary matrices
        • 1.7
          24
          Associativity Powers Nilpotent matrices
        • 1.8
          27
          Non-commutativity Left and right inverse
        • 1.9
          31
          Exercises
        • 1.10
          33
          Hints and answers
      • 37
        CHAPTER 2 Gaussian Elimination and LU -decomposition
        • 2.1
          37
          Echelon matrices Pivot elements Free and basic variables
        • 2.2
          39
          Gaussian elimination LU -decomposition
        • 2.3
          44
          Rank
        • 2.4
          48
           Exercises
        • 2.5
          50
          Hints and answers
      • 53
        CHAPTER 3 Determinants
        • 3.1
          53
          Determinant: notation and recursive definition
        • 3.2
          55
          Determinants: triangular,permutation matrices
        • 3.3
          57
          Linear and alternating properties
        • 3.4
          61
          Gaussian elimination in |A| Abstract definition
        • 3.5
          63
          The determinant of a product Expansion along an arbitrary column
        • 3.6
          65
          Even and odd permutations Complete expansion |AT|
        • 3.7
          69
          Expansion along a row The inverse matrix
        • 3.8
          72
          How to calculate the determinant Vandermonde
        • 3.9
          77
          Exercises
        • 3.10
          80
          Hints and answers
      • 83
        CHAPTER 4 Finite Dimensional Vector Spaces
        • 4.1
          83
          The definition of field
        • 4.2
          84
           The definition of vector space
        • 4.3
          85
          Linear independence Generating sets Dimension
        • 4.4
          88
           Coordinates in different bases The transition matrix
        • 4.5
          89
          The bases for a space and its subspaces
        • 4.6
          92
           Sum and intersection of subspaces Direct sum
        • 4.7
          93
          How to find a basis
        • 4.8
          100
           Exercises
        • 4.9
          102
           Hints and answers
      • 105
        CHAPTER 5 Linear Maps
        • 5.1
          105
          Definition and examples of linear maps
        • 5.2
          106
          The matrix for a linear map
        • 5.3
          110
          Matrix in a new basis Similar matrices
        • 5.4
          113
          Determinant, trace and rank of an operator Image
        • 5.5
          118
          Kernel
        • 5.6
          122
          Dimension arguments
        • 5.7
          127
          Example: some field theory
        • 5.8
          136
          Exercises
        • 5.9
          138
          Hints and answers
      • 141
        CHAPTER 6 Spectral Theory
        • 6.1
          141
          Diagonalization Eigenvectors and eigenvalues
        • 6.2
          148
           Characteristic polynomial Diagonalization
        • 6.3
          152
          Diagonalizable matrices
        • 6.4
          158
           Non-diagonalizable matrices
        • 6.5
          160
          Exercises
        • 6.6
          162
           Hints and answers
      • 165
        CHAPTER 7 The Jordan Normal Form
        • 7.1
          165
           Invariant subspaces
        • 7.2
          169
          Nilpotent operators
        • 7.3
          175
          Uniqueness Constructing the Jordan form
        • 7.4
          181
          Constructing the basis for the normal form
        • 7.5
          193
          The Jordan normal form
        • 7.6
          197
           An example of Jordanization
        • 7.7
          207
          Proving theorems using the Jordan form
        • 7.8
          211
          Exercises
        • 7.9
          213
          Hints and answers
      • 217
        CHAPTER 8 The Minimal Polynomial
        • 8.1
          217
          The Cayley-Hamilton theorem
        • 8.2
          220
          The minimal polynomial
        • 8.3
          225
          Jordan decomposition
        • 8.4
          227
           Exercises
        • 8.5
          229
          Hints and answers
      • 233
        CHAPTER 9 Functions on Matrices
        • 9.1
          233
          How to define f (A)
        • 9.2
          239
           Calculating f (A) using polynomials
        • 9.3
          241
          Lagrange interpolation
        • 9.4
          245
           Hermite interpolation
        • 9.5
          249
          Exercises
        • 9.6
          250
           Hints and answers
      • 253
        CHAPTER 10 Inequalities and Positive Matrices
        • 10.1
          253
          Inequalities and their transformations
        • 10.2
          257
          Some useful inequalities
        • 10.3
          263
          Inequalities with complex numbers
        • 10.4
          266
          Positive matrices
        • 10.5
          270
          Graphs and page ranking
        • 10.6
          273
          Exercises
        • 10.7
          274
          Hints and answers
      • 275
        CHAPTER 11 Norms
        • 11.1
          275
          Compact sets and continuous functions
        • 11.2
          279
          Norms
        • 11.3
          284
          Matrix and operator norms
        • 11.4
          290
          The condition number
        • 11.5
          292
          The spectral radius
        • 11.6
          295
          Exercises
        • 11.7
          296
          Hints and answers
      • 299
        CHAPTER 12 Inner Products and Orthogonality
        • 12.1
          299
          Hermitian conjugation
        • 12.2
          301
          Inner products and orthogonal bases
        • 12.3
          306
          Getting a norm from the inner product
        • 12.4
          308
          Unitary matrices and QR-factorization
        • 12.5
          314
          The adjoint operator
        • 12.6
          316
          Hermitian matrices
        • 12.7
          318
          Orthogonal projection and outer product
        • 12.8
          321
          Infinite dimensional vector spaces
        • 12.9
          325
          Exercises
        • 12.10
          326
          Hints and answers
      • 329
        CHAPTER 13 Singular Values
        • 13.1
          329
           Singular values
        • 13.2
          337
           Schur’s lemma
        • 13.3
          341
           Normal matrices
        • 13.4
          345
          Exercises
        • 13.5
          348
           Hints and answers
      • 353
        CHAPTER 14 Quadratic and Hermitian forms
        • 14.1
          353
           Quadratic forms and their matrices
        • 14.2
          355
          Hermitian forms
        • 14.3
          359
          Diagonalization of Hermitian forms
        • 14.4
          365
          Congruent matrices
        • 14.5
          368
          Positive definite matrices
        • 14.6
          375
          Exercises
        • 14.7
          378
          Hints and answers
      • 381
        CHAPTER 15 The Moore-Penrose Pseudoinverse
        • 15.1
          381
           Definition of the Moore-Penrose pseudoinverse
        • 15.2
          385
           Calculating the Moore-Penrose pseudoinverse
        • 15.3
          390
          The least squares method
        • 15.4
          394
          Exercises
        • 15.5
          395
          Hints and answers
      • 397
        Bibliography
      • 399
        Index
Information

Språk:

Svenska

ISBN:

9789144100968

Utgivningsår:

2014

Artikelnummer:

38585-01

Upplaga:

Första

Sidantal:

406

Köp- och leveransvillkor

Förlagskontakt

Jens Fredholm

Förläggare Tillämpad teknik

046-31 21 58 E-post
Fritjof Janson

Förlagskoordinator Kurslitteratur

046-31 22 57 E-post
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