Elementary Matrix Algebra

Name: Elementary Matrix Algebra
Brand: Dover Publications
Price: 61.49 EUR
Availability: OnlineOnly

Franz E. Hohn(Autor*in)

Dover Publications (Verlag)

Erschienen am 19. Februar 2013

1096 Seiten

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978-0-486-14372-9 (ISBN)

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Inhalt

Cover
Title Page
Copyright Page
Preface to The Third Edition
Preface to The Second Edition
Preface to The First Edition
Contents
1 Introduction to Matrix Algebra
1.1 Matrices
1.2 Equality of Matrices
1.3 Addition of Matrices
1.4 Commutative and Associative Laws of Addition
1.5 Subtraction of Matrices
1.6 Scalar Multiples of Matrices
1.7 The Multiplication of Matrices
1.8 The Properties of Matrix Multiplication
1.9 Exercises
1.10 Linear Equations in Matrix Notation
1.11 The Transpose of a Matrix
1.12 Symmetric, Skew-Symmetric, and Hermitian Matrices
1.13 Scalar Matrices
1.14 The Identity Matrix
1.15 The Inverse of a Matrix
1.16 The Product of a Row Matrix into a Column Matrix
1.17 Polynomial Functions of Matrices
1.18 Exercises
1.19 Partitioned Matrices
1.20 Exercises
2 Linear Equations
2.1 Linear Equations
2.2 Three Examples
2.3 Exercises
2.4 Equivalent Systems of Equations
2.5 The Echelon Form for Systems of Equations
2.6 Synthetic Elimination
2.7 Systems of Homogeneous Linear Equations
2.8 Exercises
2.9 Computation of the Inverse of a Matrix
2.10 Matrix Inversion by Partitioning
2.11 Exercises
2.12 Number Fields
2.13 Exercises
2.14 The General Concept of a Field
2.15 Exercises
3 Vector Geometry in E3
3.1 Geometric Representation of Vectors in E3
3.2 Operations on Vectors
3.3 Isomorphism
3.4 Length, Direction, and Sense
3.5 Orthogonality of Two Vectors
3.6 Exercises
3.7 The Vector Equation of a Line
3.8 The Vector Equation of a Plane
3.9 Exercises
3.10 Linear Combinations of Vectors in E3
3.11 Linear Dependence of Vectors
Bases
3.12 Exercises
4 Vector Geometry in n-Dimensional Space
4.1 The Real n-Space E3
4.2 Vectors in E3
4.3 Lines and Planes in E3
4.4 Linear Dependence and Independence in E3
4.5 Vector Spaces in E3
4.6 Exercises
4.7 Length and the Cauchy-Schwarz Inequality
4.8 Angles and Orthogonality in En
4.9 Half-Lines and Directed Distances
4.10 Unitary n-Space
4.11 Exercises
4.12 Linear Inequalities
4.13 Exercises
5 Vector Spaces
5.1 The General Definition of a Vector Space
5.2 Linear Combinations and Linear Dependence
5.3 Exercises
5.4 Basic Theorems on Linear Dependence
5.5 Dimension and Basis
5.6 Computation of the Dimension of a Vector Space
5.7 Exercises
5.8 Orthonormal Bases
5.9 Exercises
5.10 Intersection and Sum of Two Vector Spaces
5.11 Exercises
5.12 Isomorphic Vector Spaces
5.13 Exercises
6 The Rank of a Matrix
6.1 The Rank of a Matrix
6.2 Basic Theorems About the Rank of a Matrix
6.3 Matrix Representation of Elementary Transformations
6.4 Exercises
6.5 Homogeneous Systems of Linear Equations
6.6 Nonhomogeneous Systems of Linear Equations
6.7 Exercises
6.8 Another Look at Nonhomogeneous Systems
6.9 The Variables One Can Solve for
6.10 Basic Solutions
6.11 Exercises
7 Determinants
7.1 The Definition of a Determinant
7.2 Some Basic Theorems
7.3 The Cofactor in det A of an Element of A
7.4 Cofactors and the Computation of Determinants
7.5 Exercises
7.6 The Determinant of the Product of Two Matrices
7.7 A Formula for A-1
7.8 Determinants and the Rank of a Matrix
7.9 Solution of Systems of Equations by Using Determinants
7.10 A Geometrical Application of Determinants
7.11 Exercises
7.12 More About the Rank of a Matrix
7.13 Definitions
7.14 The Laplace Expansion
7.15 The Determinant of a Product of Two Square Matrices
7.16 The Adjoint Matrix
7.17 The Row-and-Column Expansion
7.18 The Diagonal Expansion of the Determinant of a Matrix
7.19 Exercises
8 Linear Transformations
8.1 Mappings
8.2 Linear Mappings
8.3 Some Properties of Linear Operators on Vector Spaces
8.4 Exercises
8.5 Linear Transformations of Coordinates
8.6 Transformation of a Linear Operator
8.7 Exercises
8.8 The Algebra of Linear Operators
8.9 Groups of Operators
8.10 Exercises
8.11 Unitary and Orthogonal Matrices
8.12 Exercises
8.13 Unitary Transformations
8.14 Orthogonal Transformations
8.15 The Eulerian Angles
8.16 The Triangularization of a Real Matrix
8.17 Exercises
8.18 Orthogonal Vector Spaces
8.19 Exercises
8.20 Projections
8.21 Orthogonal Projections in un
8.22 Exercises
9 The Characteristic Value Problem
9.1 Definition of the Characteristic Value Problem
9.2 Four Examples
9.3 Two Basic Theorems
9.4 Exercises
9.5 The Characteristic Polynomial and Its Roots
9.6 Similar Matrices
9.7 Exercises
9.8 The Characteristic Roots of a Hermitian Matrix
9.9 The Diagonal Form of a Hermitian Matrix
9.10 The Diagonalization of a Hermitian Matrix
9.11 Examples
9.12 Triangularization of an Arbitrary Matrix
9.13 Normal Matrices
9.14 Exercises
9.15 Characteristic Roots of a Polynomial Function of a Matrix
9.16 The Cayley-Hamilton Theorem
9.17 The Minimum Polynomial of a Matrix
9.18 Powers of Matrices
9.19 Exercises
10 Quadratic, Bilinear, and Hermitian Forms
10.1 Quadratic Forms
10.2 Diagonalization of Quadratic Forms
10.3 A Geometrical Application
10.4 Definite Forms and Matrices
10.5 Exercises
10.6 Lagrange's Reduction
10.7 Kronecker's Reduction
10.8 Sylvester's Law of Inertia for Real Quadratic Forms
10.9 A Necessary and Sufficient Condition for Positive Definiteness
10.10 An Important Example
10.11 Exercises
10.12 Pairs of Quadratic Forms
10.13 Values of Quadratic Forms
10.14 Exercises
10.15 Bilinear Forms
10.16 The Equivalence of Bilinear Forms
10.17 Cogredient and Contragredient Transformations
10.18 Exercises
10.19 Hermitian Forms
10.20 Definite Hermitian Forms
10.21 Exercises
I The Notations S and p
II The Algebra of Complex Numbers
Bibliography
Index

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