This self-contained textbook takes a matrix-oriented approach to linear algebra and presents a complete theory, including all details and proofs, culminating in the Jordan canonical form and its proof. Throughout the development, the applicability of the results is highlighted. Additionally, the book presents special topics from applied linear algebra including matrix functions, the singular value decomposition, the Kronecker product and linear matrix equations.
The matrix-oriented approach to linear algebra leads to a better intuition and a deeper understanding of the abstract concepts, and therefore simplifies their use in real world applications. Some of these applications are presented in detailed examples. In several ‘MATLAB-Minutes’ students can comprehend the concepts and results using computational experiments. Necessary basics for the use of MATLAB are presented in a short introduction. Students can also actively work with the material and practice their mathematical skills in more than 300 exercises.
Author(s): Jörg Liesen, Volker Mehrmann
Series: Springer Undergraduate Mathematics Series
Edition: 1st ed. 2015
Publisher: Springer
Year: 2015
Language: English
Pages: 335
Preface
Preface to the First German Edition
Contents
1 Linear Algebra in Every Day Life
1.1 The PageRank Algorithm
1.2 No Claim Discounting in Car Insurances
1.3 Production Planning in a Plant
1.4 Predicting Future Profits
1.5 Circuit Simulation
2 Basic Mathematical Concepts
2.1 Sets and Mathematical Logic
2.2 Maps
2.3 Relations
3 Algebraic Structures
3.1 Groups
3.2 Rings and Fields
4 Matrices
4.1 Basic Definitions and Operations
4.2 Matrix Groups and Rings
5 The Echelon Form and the Rank of Matrices
5.1 Elementary Matrices
5.2 The Echelon Form and Gaussian Elimination
5.3 Rank and Equivalence of Matrices
6 Linear Systems of Equations
7 Determinants of Matrices
7.1 Definition of the Determinant
7.2 Properties of the Determinant
7.3 Minors and the Laplace Expansion
8 The Characteristic Polynomial and Eigenvalues of Matrices
8.1 The Characteristic Polynomial and the Cayley-Hamilton Theorem
8.2 Eigenvalues and Eigenvectors
8.3 Eigenvectors of Stochastic Matrices
9 Vector Spaces
9.1 Basic Definitions and Properties of Vector Spaces
9.2 Bases and Dimension of Vector Spaces
9.3 Coordinates and Changes of the Basis
9.4 Relations Between Vector Spaces and Their Dimensions
10 Linear Maps
10.1 Basic Definitions and Properties of Linear Maps
10.2 Linear Maps and Matrices
11 Linear Forms and Bilinear Forms
11.1 Linear Forms and Dual Spaces
11.2 Bilinear Forms
11.3 Sesquilinear Forms
12 Euclidean and Unitary Vector Spaces
12.1 Scalar Products and Norms
12.2 Orthogonality
12.3 The Vector Product in mathbbR3,1
13 Adjoints of Linear Maps
13.1 Basic Definitions and Properties
13.2 Adjoint Endomorphisms and Matrices
14 Eigenvalues of Endomorphisms
14.1 Basic Definitions and Properties
14.2 Diagonalizability
14.3 Triangulation and Schur's Theorem
15 Polynomials and the Fundamental Theorem of Algebra
15.1 Polynomials
15.2 The Fundamental Theorem of Algebra
16 Cyclic Subspaces, Duality and the Jordan Canonical Form
16.1 Cyclic f-invariant Subspaces and Duality
16.2 The Jordan Canonical Form
16.3 Computation of the Jordan Canonical Form
17 Matrix Functions and Systems of Differential Equations
17.1 Matrix Functions and the Matrix Exponential Function
17.2 Systems of Linear Ordinary Differential Equations
18 Special Classes of Endomorphisms
18.1 Normal Endomorphisms
18.2 Orthogonal and Unitary Endomorphisms
18.3 Selfadjoint Endomorphisms
19 The Singular Value Decomposition
20 The Kronecker Product and Linear Matrix Equations
Appendix A
A Short Introduction to MATLAB
Selected HistoricalWorks on Linear Algebra
Bibliography
Index
