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THE GEOMETRY OF LINEAR ALGEBRA
Corinne A. Manogue, Tevian Dray
Contents
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Contents
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Front Matter
Colophon
1
Complex Numbers
The Complex Plane
Complex Conjugate and Norm
Algebra with Complex Numbers: Rectangular Form
Division: Rectangular Form
Euler's Formula
Exponential Form
Roots of Complex Numbers
Logarithms of Complex Numbers
2
Operations with Matrices
Matrix Addition
Scalar Multiplication
Matrix Multiplication
Transpose
Hermitian Adjoint
Dot Products
Inner Products for Complex Vectors
Trace
Determinants
Inverses
3
Eigenvectors and Eigenvalues
What are Eigenvectors?
Finding Eigenvalues
Finding Eigenvectors
Normalization of Eigenvectors
Diagonal Matrices
Degeneracy
Using Eigenvectors as a Natural Basis
4
Special Matrices
Hermitian Matrices
Properties of Hermitian Matrices
Commuting Matrices
Properties of Unitary Matrices
Unitary Matrices
Change of Basis
Symmetry Operations
Matrix Examples
Matrix Decompositions
Matrix Exponentials
Evolution Equation
5
Vector Spaces
Definition of a Vector Space
Definition and Properties of an Inner Product
Linear Operators
6
Delta Functions
Step Functions
The Dirac Delta Function
Properties of the Dirac Delta Function
Representations of the Dirac Delta Function
The Dirac Delta Function in Three Dimensions
The Exponential Representation of the Dirac Delta Function
7
Power Series
Power Series
Dimensions in Power Series
Approximations using Power Series
Visualization of Power Series
Common Power Series
Convergence of Power Series
Theorems about Power Series
8
Differential Equations
Definitions and Theorems
First Order ODEs: Notation and Theorems
Separable ODEs
Exact ODEs
The word “Linear”: Definitions and Theorems
Theorems about Linear ODEs
Constant Coefficients, Homogeneous
Linear Independence
Constant Coefficients, Inhomogeneous
Power Series Solutions, Theorems
Power Series Solutions: Method/Example
9
Fourier Series
Fourier Series Motivation
Sums of Harmonic Functions
Products of Harmonic Functions
Overlap Integrals
Finding Coefficients
Fourier Series Example
The Gibbs Phenomenon
Completeness
Symmetries
10
Fourier Transforms
Definition of Fourier Transform
Examples of Fourier Transforms
Fourier Uncertainties
Wave Packets
11
Partial Differential Equations
Important PDEs in Physics
Classification of PDEs
PDE Theorems
Separation of Variables
Sturm–Liouville Theory
12
Special Functions
Back Matter
Bibliography
Authored in PreTeXt
Chapter
4
Special Matrices
¶
4.1
Hermitian Matrices
4.2
Properties of Hermitian Matrices
4.3
Commuting Matrices
4.4
Properties of Unitary Matrices
4.5
Unitary Matrices
4.6
Change of Basis
4.7
Symmetry Operations
4.8
Matrix Examples
4.9
Matrix Decompositions
4.10
Matrix Exponentials
4.11
Evolution Equation
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