Linear Algebra and Group Theory for Physicists and Engineers

Preface
How to Use the Book in Academic Courses?
Roadmaps: How to Read the Book?
Contents
Part I Introduction to Linear Algebra
1 Vectors and Matrices
1.1 Vectors in Two and Three Dimensions
1.1.1 Two-Dimensional Vectors
1.1.2 Adding Vectors
1.1.3 Scalar Times Vector
1.1.4 Three-Dimensional Vectors
1.2 Vectors in Higher Dimensions
1.2.1 Multidimensional Vectors
1.2.2 Associative Law
1.2.3 The Origin
1.2.4 Multiplication and Its Laws
1.2.5 Distributive Laws
1.3 Complex Numbers and Vectors
1.3.1 Complex Numbers
1.3.2 Complex Vectors
1.4 Rectangular Matrix
1.4.1 Matrices
1.4.2 Adding Matrices
1.4.3 Scalar Times Matrix
1.4.4 Matrix Times Vector
1.4.5 Matrix-Times-Matrix
1.4.6 Distributive and Associative Laws
1.4.7 The Transpose Matrix
1.5 Square Matrix
1.5.1 Symmetric Square Matrix
1.5.2 The Identity Matrix
1.5.3 The Inverse Matrix as a Mapping
1.5.4 Inverse and Transpose
1.6 Complex Matrix and Its Hermitian Adjoint
1.6.1 The Hermitian Adjoint
1.6.2 Hermitian (Self-Adjoint) Matrix
1.7 Inner Product and Norm
1.7.1 Inner (Scalar) Product
1.7.2 Bilinearity
1.7.3 Skew-Symmetry
1.7.4 Norm
1.7.5 Normalization
1.7.6 Other Norms
1.7.7 Inner Product and the Hermitian Adjoint
1.7.8 Inner Product and a Hermitian Matrix
1.8 Orthogonal and Unitary Matrix
1.8.1 Inner Product of Column Vectors
1.8.2 Orthogonal and Orthonormal Column Vectors
1.8.3 Projection Matrix and Its Null Space
1.8.4 Unitary and Orthogonal Matrix
1.9 Eigenvalues and Eigenvectors
1.9.1 Eigenvectors and Their Eigenvalues
1.9.2 Singular Matrix and Its Null Space
1.9.3 Eigenvalues of the Hermitian Adjoint
1.9.4 Eigenvalues of a Hermitian Matrix
1.9.5 Eigenvectors of a Hermitian Matrix
1.10 The Sine Transform
1.10.1 Discrete Sine Waves
1.10.2 Orthogonality of the Discrete Sine Waves
1.10.3 The Sine Transform
1.10.4 Diagonalization
1.10.5 Sine Decomposition
1.10.6 Multiscale Decomposition
1.11 The Cosine Transform
1.11.1 Discrete Cosine Waves
1.11.2 Orthogonality of the Discrete Cosine Waves
1.11.3 The Cosine Transform
1.11.4 Diagonalization
1.11.5 Cosine Decomposition
1.12 Positive (Semi)definite Matrix
1.12.1 Positive Semidefinite Matrix
1.12.2 Positive Definite Matrix
1.13 Exercises: Generalized Eigenvalues
1.13.1 The Cauchy–Schwarz Inequality
1.13.2 The Triangle Inequality
1.13.3 Generalized Eigenvalues
1.13.4 Root of Unity and Fourier Transform
2 Determinant and Vector Product and Their Applications in Geometrical Mechanics
2.1 The Determinant
2.1.1 Minors and the Determinant
2.1.2 Examples
2.1.3 Algebraic Properties
2.1.4 The Inverse Matrix in Its Explicit Form
2.1.5 Cramer's Rule
2.2 Vector (Cross) Product
2.2.1 Standard Unit Vectors in 3-D
2.2.2 Inner Product—Orthogonal Projection
2.2.3 Vector (Cross) Product
2.2.4 The Right-Hand Rule
2.3 Orthogonalization
2.3.1 Invariance Under Orthogonal Transformation
2.3.2 Relative Axis System: Gram–Schmidt Process
2.3.3 Angle Between Vectors
2.4 Linear and Angular Momentum
2.4.1 Linear Momentum
2.4.2 Radial Component: Orthogonal Projection
2.4.3 Angular Momentum
2.4.4 Angular Momentum and Its Norm
2.4.5 Linear Momentum and Its Nonradial Component
2.4.6 Linear Momentum and Its Orthogonal Decomposition
2.5 Angular Velocity
2.5.1 Angular Velocity
2.5.2 The Rotating Axis System
2.5.3 Velocity and Its Decomposition
2.6 Real and Fictitious Forces
2.6.1 The Centrifugal Force
2.6.2 The Centripetal Force
2.6.3 The Euler Force
2.6.4 The Earth and Its Rotation
2.6.5 Coriolis Force
2.7 Exercises: Inertia and Principal Axes
2.7.1 Rotation and Euler Angles
2.7.2 Algebraic Right-Hand Rule
2.7.3 Linear Momentum and Its Conservation
2.7.4 Principal Axes
2.7.5 The Inertia Matrix
2.7.6 The Triple Vector Product
2.7.7 Linear Momentum: Orthogonal Decomposition
2.7.8 The Centrifugal and Centripetal Forces
2.7.9 The Inertia Matrix Times the Angular Velocity
2.7.10 Angular Momentum and Its Conservation
2.7.11 Rigid Body
2.7.12 The Percussion Point
2.7.13 Bohr's Atom and Energy Levels
3 Markov Matrix and Its Spectrum: Toward Search Engines
3.1 Characteristic Polynomial and Spectrum
3.1.1 Null Space and Characteristic Polynomial
3.1.2 Spectrum and Spectral Radius
3.2 Graph and Its Matrix
3.2.1 Weighted Graph
3.2.2 Markov Matrix
3.2.3 Example: Uniform Probability
3.3 Flow and Mass
3.3.1 Stochastic Flow: From State to State
3.3.2 Mass Conservation
3.4 The Steady State
3.4.1 The Spectrum of Markov Matrix
3.4.2 Converging Markov Chain
3.4.3 The Steady State
3.4.4 Search Engine in the Internet
3.5 Exercises: Gersgorin's Theorem
3.5.1 Gersgorin's Theorem
4 Special Relativity: Algebraic Point of View
4.1 Adding Velocities (or Speeds)
4.1.1 How to Add Velocities?
4.1.2 Einstein's Law: Never Exceed the Speed of Light!
4.1.3 Particle as Fast as Light
4.1.4 Singularity: Indistinguishable Particles
4.2 Systems and Their Time
4.2.1 Inertial Reference Frame
4.2.2 How to Measure Time?
4.2.3 The Self-system
4.2.4 Synchronization
4.3 Lorentz Group of Transformations (Matrices)
4.3.1 Space and Time: Same Status
4.3.2 Lorentz Transformation
4.3.3 Lorentz Matrix and the Infinity Point
4.3.4 Interchanging Coordinates
4.3.5 Composite Transformation
4.3.6 The Inverse Transformation
4.3.7 Abelian Group of Lorentz Matrices
4.4 Proper Time in the Self-system
4.4.1 Proper Time: Invariant
4.4.2 Time Dilation
4.4.3 Length Contraction
4.4.4 Simultaneous Events
4.5 Spacetime and Velocity
4.5.1 Doppler's Effect
4.5.2 Velocity in Spacetime
4.5.3 Moebius Transformation
4.5.4 Perpendicular Velocity
4.6 Relativistic Momentum and its Conservation
4.6.1 Invariant Mass
4.6.2 Momentum: Old Definition
4.6.3 Relativistic Momentum
4.6.4 Rest Mass vs. Relativistic Mass
4.6.5 Moderate (Nonrelativistic) Velocity
4.6.6 Closed System: Lose Mass—Gain Motion
4.6.7 The Momentum Matrix
4.6.8 Momentum and its Conservation
4.7 Relativistic Energy and its Conservation
4.7.1 Force: Derivative of Momentum
4.7.2 Open System: Constant Mass
4.7.3 Relativistic Energy: Kinetic Plus Potential
4.7.4 Moderate (Nonrelativistic) Velocity
4.8 Mass and Energy: Closed vs. Open System
4.8.1 Why Is It Called Rest Mass?
4.8.2 Mass is Invariant
4.8.3 Energy is Conserved—Mass Is Not
4.8.4 Particle Starting to Move
4.8.5 Say Mass, Not Rest Mass
4.8.6 Decreasing Mass in the Lab
4.8.7 Closed System: Energy Can Only Convert
4.8.8 Open System
4.8.9 Mass in a Closed System
4.9 Momentum–Energy and Their Transformation
4.9.1 New Mass
4.9.2 Spacetime
4.9.3 A Naive Approach
4.9.4 The Momentum–Energy Vector
4.9.5 The Momentum Matrix in Spacetime
4.9.6 Lorentz Transformation on Momentum–Energy
4.10 Energy and Mass
4.10.1 Invariant Nuclear Energy
4.10.2 Invariant Mass
4.10.3 Einstein's Formula
4.11 Center of Mass
4.11.1 Collection of Subparticles
4.11.2 Center of Mass
4.11.3 The Mass of the Collection
4.12 Oblique Force and Momentum
4.12.1 Oblique Momentum in x'-y'
4.12.2 View from Spacetime
4.12.3 The Lab: The New Self-system
4.13 Force in an Open System
4.13.1 Force in an Open Passive System
4.13.2 What Is the Force in Spacetime?
4.13.3 Proper Time in the Lab
4.13.4 Nearly Proper Time in the Lab
4.14 Perpendicular Force
4.14.1 Force: Time Derivative of Momentum
4.14.2 Passive System—Strong Perpendicular Force
4.15 Nonperpendicular Force
4.15.1 Force: Time Derivative of Momentum
4.15.2 Energy in an Open System
4.15.3 Open System—Constant Mass
4.15.4 Nearly Constant Energy in the Lab
4.15.5 Nonperpendicular Force: Same at All Systems
4.15.6 The Photon Paradox
4.16 Exercises: Special Relativity in 3-D
4.16.1 Lorentz Matrix and its Determinant
4.16.2 Motion in 3-D
Part II Introduction to Group Theory
5 Groups and Isomorphism Theorems
5.1 Moebius Transformation and Matrix
5.1.1 Riemann Sphere—Extended Complex Plane
5.1.2 Moebius Transformation and the Infinity Point
5.1.3 The Inverse Transformation
5.1.4 Moebius Transformation as a Matrix
5.1.5 Product of Moebius Transformations
5.2 Matrix: A Function
5.2.1 Matrix as a Vector Function
5.2.2 Matrix Multiplication as Composition
5.3 Group and its Properties
5.3.1 Group
5.3.2 The Unit Element
5.3.3 Inverse Element
5.4 Mapping and Homomorphism
5.4.1 Mapping and its Origin
5.4.2 Homomorphism
5.4.3 Mapping the Unit Element
5.4.4 Preserving the Inverse Operation
5.4.5 Kernel of a Mapping
5.5 The Center and Kernel Subgroups
5.5.1 Subgroup
5.5.2 The Center Subgroup
5.5.3 The Kernel Subgroup
5.6 Equivalence Classes
5.6.1 Equivalence Relation in a Set
5.6.2 Decomposition into Equivalence Classes
5.6.3 Family of Equivalence Classes
5.6.4 Equivalence Relation Induced by a Subgroup
5.6.5 Equivalence Classes Induced by a Subgroup
5.7 The Factor Group
5.7.1 The New Set G/S
5.7.2 Normal Subgroup
5.7.3 The Factor (Quotient) Group
5.7.4 Is the Kernel Normal?
5.7.5 Isomorphism on the Factor Group
5.7.6 The Fundamental Theorem of Homomorphism
5.8 Geometrical Applications
5.8.1 Application in Moebius Transformations
5.8.2 Two-Dimensional Vector Set
5.8.3 Geometrical Decomposition into Planes
5.8.4 Family of Planes
5.8.5 Action of Factor Group
5.8.6 Composition of Functions
5.8.7 Oblique Projection: Extended Cotangent
5.8.8 Homomorphism onto Moebius Transformations
5.8.9 The Kernel
5.8.10 Eigenvectors and Fixed Points
5.8.11 Isomorphism onto Moebius Transformations
5.9 Application in Continued Fractions
5.9.1 Continued Fractions
5.9.2 Algebraic Formulation
5.9.3 The Approximants
5.9.4 Algebraic Convergence
5.10 Isomorphism Theorems
5.10.1 The Second Isomorphism Theorem
5.10.2 The Third Isomorphism Theorem
5.11 Exercises
6 Projective Geometry with Applications in Computer Graphics
6.1 Circles and Spheres
6.1.1 Degenerate ``Circle''
6.1.2 Antipodal Points in the Unit Circle
6.1.3 More Circles
6.1.4 Antipodal Points in the Unit Sphere
6.1.5 General Multidimensional Hypersphere
6.1.6 Complex Coordinates
6.2 The Complex Projective Plane
6.2.1 The Complex Projective Plane
6.2.2 Topological Homeomorphism onto the Sphere
6.2.3 The Center and its Subgroups
6.2.4 Group Product
6.2.5 The Center—a Group Product
6.2.6 How to Divide by a Product?
6.2.7 How to Divide by a Circle?
6.2.8 Second and Third Isomorphism Theorems
6.3 The Real Projective Line
6.3.1 The Real Projective Line
6.3.2 The Divided Circle
6.4 The Real Projective Plane
6.4.1 The Real Projective Plane
6.4.2 Oblique Projection
6.4.3 Radial Projection
6.4.4 The Divided Sphere
6.4.5 Infinity Points
6.4.6 The Infinity Circle
6.4.7 Lines as Level Sets
6.5 Infinity Points and Line
6.5.1 Infinity Points and their Projection
6.5.2 Riemannian Geometry
6.5.3 A Joint Infinity Point
6.5.4 Two Lines Share a Unique Point
6.5.5 Parallel Lines Do Meet
6.5.6 The Infinity Line
6.5.7 Duality: Two Points Make a Unique Line
6.6 Conics and Envelopes
6.6.1 Conic as a Level Set
6.6.2 New Axis System
6.6.3 The Projected Conic
6.6.4 Ellipse, Hyperbola, or Parabola
6.6.5 Tangent Planes
6.6.6 Envelope
6.6.7 The Inverse Mapping
6.7 Duality: Conic–Envelope
6.7.1 Conic and its Envelope
6.7.2 Hyperboloid and its Projection
6.7.3 Projective Mappings
6.8 Applications in Computer Graphics
6.8.1 Translation
6.8.2 Motion in a Curved Trajectory
6.8.3 The Translation Matrix
6.8.4 General Translation of a Planar Object
6.8.5 Unavailable Tangent
6.8.6 Rotation
6.8.7 Relation to the Complex Projective Plane
6.9 The Real Projective Space
6.9.1 The Real Projective Space
6.9.2 Oblique Projection
6.9.3 Radial Projection
6.10 Duality: Point–Plane
6.10.1 Points and Planes
6.10.2 The Extended Vector Product
6.10.3 Three Points Make a Unique Plane
6.10.4 Three Planes Share a Unique Point
6.11 Exercises
7 Quantum Mechanics: Algebraic Point of View
7.1 Nondeterminism
7.1.1 Relativistic Observation
7.1.2 Determinism
7.1.3 Nondeterminism and Observables
7.2 State: Wave Function
7.2.1 Physical State
7.2.2 The Diagonal Position Matrix
7.2.3 Normalization
7.2.4 State and Its Overall Phase
7.2.5 Dynamics: Schrodinger Picture
7.2.6 Wave Function and Phase
7.2.7 Phase and Interference
7.3 Observables: Which Is First?
7.3.1 Measurement: The State Is Gone
7.3.2 The Momentum Matrix and Its Eigenvalues
7.3.3 Ordering Matters!
7.3.4 Commutator
7.3.5 Planck Constant
7.4 Observable and Its Expectation
7.4.1 Observable (Measurable)
7.4.2 Hermitian and Anti-Hermitian Parts
7.4.3 Symmetrization
7.4.4 Observation
7.4.5 Random Variable
7.4.6 Observable and Its Expectation
7.5 Heisenberg's Uncertainty Principle
7.5.1 Variance
7.5.2 Covariance
7.5.3 Heisenberg's Uncertainty Principle
7.6 Wave: Debroglie Relation
7.6.1 Infinite Matrix (or Operator)
7.6.2 Momentum: Another Operator
7.6.3 The Commutator
7.6.4 Wave: An Eigenfunction
7.6.5 Duality: Particle—Matter or Wave?
7.6.6 Debroglie's Relation: Momentum–Wave Number
7.7 Planck and Schrodinger Equations
7.7.1 Hamiltonian: Energy Operator
7.7.2 Time–Energy Uncertainty
7.7.3 Planck Relation: Frequency–Energy
7.7.4 No Potential: Momentum Is Conserved Too
7.7.5 Stability in Bohr's Atom
7.8 Eigenvalues
7.8.1 Shifting an Eigenvalue
7.8.2 Shifting an Eigenvalue of a Product
7.8.3 A Number Operator
7.8.4 Eigenvalue—Expectation
7.8.5 Down the Ladder
7.8.6 Null Space
7.8.7 Up the Ladder
7.9 Hamiltonian
7.9.1 Harmonic Oscillator
7.9.2 Concrete Number Operator
7.9.3 Energy Levels
7.9.4 Ground State (Zero-Point Energy)
7.9.5 Gaussian Distribution
7.10 Coherent State
7.10.1 Energy Levels and Their Superposition
7.10.2 Energy Levels and Their Precession
7.10.3 Coherent State
7.10.4 Probability to Have Certain Energy
7.10.5 Poisson Distribution
7.10.6 Conservation of Energy
7.11 Particle in 3-D
7.11.1 The Discrete 2-D Grid
7.11.2 Position and Momentum
7.11.3 Tensor Product
7.11.4 Commutativity
7.11.5 3-D Grid
7.11.6 Bigger Tensor Product
7.12 Angular Momentum
7.12.1 Angular Momentum Component
7.12.2 Using the Commutator
7.12.3 Up the Ladder
7.12.4 Down the Ladder
7.12.5 Angular Momentum
7.13 Toward the Path Integral
7.13.1 What Is an Electron?
7.13.2 Dynamics
7.13.3 Reversibility
7.13.4 Toward Spin
7.14 Exercises: Spin
7.14.1 Eigenvalues and Eigenvectors
7.14.2 Hamiltonian and Energy Levels
7.14.3 The Ground State and Its Conservation
7.14.4 Coherent State and Its Dynamics
7.14.5 Entanglement
7.14.6 Angular Momentum and Its Eigenvalues
7.14.7 Spin-One
7.14.8 Spin-One-Half and Pauli Matrices
7.14.9 Polarization
7.14.10 Conjugation
7.14.11 Dirac Matrices Anti-commute
7.14.12 Dirac Matrices in Particle Physics
Part III Polynomials and Basis Functions
8 Polynomials and Their Gradient
8.1 Polynomials and Their Arithmetic Operations
8.1.1 Polynomial of One Variable
8.1.2 Real vs. Complex Polynomial
8.1.3 Addition
8.1.4 Scalar Multiplication
8.1.5 Multiplying Polynomials: Convolution
8.1.6 Example: Scalar Multiplication
8.2 Polynomial and Its Value
8.2.1 Value at a Given Point
8.2.2 The Naive Method
8.2.3 Using the Distributive Law
8.2.4 Recursion: Horner's Algorithm
8.2.5 Complexity: Mathematical Induction
8.3 Composition
8.3.1 Mathematical Induction
8.3.2 The Induction Step
8.3.3 Recursion: A New Horner Algorithm
8.4 Natural Number as a Polynomial
8.4.1 Decimal Polynomial
8.4.2 Binary Polynomial
8.5 Monomial and Its Value
8.5.1 Monomial
8.5.2 A Naive Method
8.5.3 Horner Algorithm: Implicit Form
8.5.4 Mathematical Induction
8.5.5 The Induction Step
8.5.6 Complexity: Total Cost
8.5.7 Recursion Formula
8.6 Differentiation
8.6.1 Derivative of a Polynomial
8.6.2 Second Derivative
8.6.3 High-Order Derivatives
8.7 Integration
8.7.1 Indefinite Integral
8.7.2 Definite Integral over an Interval
8.7.3 Examples
8.7.4 Definite Integral over the Unit Interval
8.8 Sparse Polynomials
8.8.1 Sparse Polynomial
8.8.2 Sparse Polynomial: Explicit Form
8.8.3 Sparse Polynomial: Recursive Form
8.8.4 Improved Horner Algorithm
8.8.5 Power of a Polynomial
8.8.6 Composition
8.9 Polynomial of Two Variables
8.9.1 Polynomial of Two Independent Variables
8.9.2 Arithmetic Operations
8.10 Differentiation and Integration
8.10.1 Partial Derivatives
8.10.2 The Gradient
8.10.3 Integral over the Unit Triangle
8.10.4 Second Partial Derivatives
8.10.5 Degree
8.11 Polynomial of Three Variables
8.11.1 Polynomial of Three Independent Variables
8.12 Differentiation and Integration
8.12.1 Partial Derivatives
8.12.2 The Gradient
8.12.3 Vector Field (or Function)
8.12.4 The Jacobian
8.12.5 Integral over the Unit Tetrahedron
8.13 Normal and Tangential Derivatives
8.13.1 Directional Derivative
8.13.2 Normal Derivative
8.13.3 Differential Operator
8.13.4 High-Order Normal Derivatives
8.13.5 Tangential Derivative
8.14 High-Order Partial Derivatives
8.14.1 High-Order Partial Derivatives
8.14.2 The Hessian
8.14.3 Degree
8.15 Exercises: Convolution
8.15.1 Convolution and Polynomials
8.15.2 Polar Decomposition
9 Basis Functions: Barycentric Coordinates in 3-D
9.1 Tetrahedron and its Mapping
9.1.1 General Tetrahedron
9.1.2 Integral Over a Tetrahedron
9.1.3 The Chain Rule
9.1.4 Degrees of Freedom
9.2 Barycentric Coordinates in 3-D
9.2.1 Barycentric Coordinates in 3-D
9.2.2 The Inverse Mapping
9.2.3 Geometrical Interpretation
9.2.4 The Chain Rule and Leibniz Rule
9.2.5 Integration in Barycentric Coordinates
9.3 Independent Degrees of Freedom
9.3.1 Continuity Across an Edge
9.3.2 Smoothness Across an Edge
9.3.3 Continuity Across a Side
9.3.4 Independent Degrees of Freedom
9.4 Piecewise-Polynomial Functions
9.4.1 Smooth Piecewise-Polynomial Function
9.4.2 Continuous Piecewise-Polynomial Function
9.5 Basis Functions
9.5.1 Side Midpoint Basis Function
9.5.2 Edge-Midpoint Basis Function
9.5.3 Hessian-Related Corner Basis Function
9.5.4 Gradient-Related Corner Basis Function
9.5.5 Corner Basis Function
9.6 Numerical Experiment: Electromagnetic Waves
9.6.1 Frequency and Wave Number
9.6.2 Adaptive Mesh Refinement
9.7 Numerical Results
9.7.1 High-Order Finite Elements
9.7.2 Linear Adaptive Finite Elements
9.8 Exercises
Part IV Finite Elements in 3-D
10 Automatic Mesh Generation
10.1 The Refinement Step
10.1.1 Iterative Multilevel Refinement
10.1.2 Conformity
10.1.3 Regular Mesh
10.1.4 How to Preserve Regularity?
10.2 Approximating a 3-D Domain
10.2.1 Implicit Domain
10.2.2 Example: A Nonconvex Domain
10.2.3 How to Find a Boundary Point?
10.3 Approximating a Convex Boundary
10.3.1 Boundary Refinement
10.3.2 Boundary Edge and Triangle
10.3.3 How to Fill a Valley?
10.3.4 How to Find a Boundary Edge?
10.3.5 Locally Convex Boundary: Gram–Schmidt Process
10.4 Approximating a Nonconvex Domain
10.4.1 Locally Concave Boundary
10.4.2 Convex Meshes
10.5 Exercises
11 Mesh Regularity
11.1 Angle and Sine in 3-D
11.1.1 Sine in a Tetrahedron
11.1.2 Minimal Angle
11.1.3 Proportional Sine
11.1.4 Minimal Sine
11.2 Adequate Equivalence
11.2.1 Equivalent Regularity Estimates
11.2.2 Inadequate Equivalence
11.2.3 Ball Ratio
11.3 Numerical Experiment
11.3.1 Mesh Regularity
11.3.2 Numerical Results
11.4 Exercises
12 Numerical Integration
12.1 Integration in 3-D
12.1.1 Volume of a Tetrahedron
12.1.2 Integral in 3-D
12.1.3 Singularity
12.2 Changing Variables
12.2.1 Spherical Coordinates
12.2.2 Partial Derivatives
12.2.3 The Jacobian
12.2.4 Determinant of Jacobian
12.2.5 Integrating a Composite Function
12.3 Integration in the Meshes
12.3.1 Integrating in a Ball
12.3.2 Stopping Criterion
12.3.3 Richardson Extrapolation
12.4 Exercises
13 Spline: Variational Model in 3-D
13.1 Expansion in Basis Functions
13.1.1 Degrees of Freedom in the Mesh
13.1.2 The Function Space and Its Basis
13.2 The Stiffness Matrix
13.2.1 Assemble the Stiffness Matrix
13.2.2 How to Order the Basis Functions?
13.3 Finding the Optimal Spline
13.3.1 Minimum Energy
13.3.2 The Schur Complement
13.4 Exercises
Part V Permutation Group and Determinant in Quantum Chemistry
14 Permutation Group and the Determinant
14.1 Permutation
14.1.1 Permutation
14.1.2 Switch
14.1.3 Cycle
14.2 Decomposition
14.2.1 Composition (Product)
14.2.2 3-Cycle
14.2.3 4-Cycle
14.2.4 How to Decompose a Permutation?
14.3 Permutations and Their Group
14.3.1 Group of Permutations
14.3.2 How Many Permutations?
14.4 Determinant
14.4.1 Determinant: A New Definition
14.4.2 Determinant of the Transpose
14.4.3 Determinant of a Product
14.4.4 Orthogonal Matrix
14.4.5 Unitary Matrix
14.5 The Characteristic Polynomial
14.5.1 The Characteristic Polynomial
14.5.2 Trace—Sum of Eigenvalues
14.5.3 Determinant—Product of Eigenvalues
14.6 Exercises: Permutation and Its Structure
14.6.1 Decompose as a Product of Switches
15 Electronic Structure in the Atom: The Hartree–Fock System
15.1 Wave Function
15.1.1 Particle and Its Wave Function
15.1.2 Entangled Particles
15.1.3 Disentangled Particles
15.2 Electrons in Their Orbitals
15.2.1 Atom: Electrons in Orbitals
15.2.2 Potential Energy and Its Expectation
15.3 Distinguishable Electrons
15.3.1 Hartree Product
15.3.2 Potential Energy of the Hartree Product
15.4 Indistinguishable Electrons
15.4.1 Indistinguishable Electrons
15.4.2 Pauli's Exclusion Principle: Slater Determinant
15.5 Orbitals and Their Canonical Form
15.5.1 The Overlap Matrix and Its Diagonal Form
15.5.2 Unitary Transformation
15.5.3 Orthogonal Orbitals
15.5.4 Slater Determinant and Unitary Transformation
15.5.5 Orthonormal Orbitals: The Canonical Form
15.5.6 Slater Determinant and Its Overlap
15.6 Expected Energy
15.6.1 Coulomb and Exchange Integrals
15.6.2 Effective Potential Energy
15.6.3 Kinetic Energy
15.6.4 The Schrodinger Equation in Its Integral Form
15.7 The Hartree–Fock System
15.7.1 Basis Functions and the Coefficient Matrix
15.7.2 The Mass Matrix
15.7.3 The Pseudo-Eigenvalue Problem
15.7.4 Is the Canonical Form Plausible?
15.8 Exercises: Electrostatic Potential
15.8.1 Potential: Divergence of Flux
Part VI The Jordan Form
16 The Jordan Form
16.1 Nilpotent Matrix and Generalized Eigenvectors
16.1.1 Nilpotent Matrix
16.1.2 Cycle and Invariant Subspace
16.1.3 Generalized Eigenvectors and Their Linear Independence
16.1.4 More General Cases
16.1.5 Linear Dependence
16.1.6 More General Cases
16.2 Nilpotent Matrix and Its Jordan Form
16.2.1 How to Design a Jordan Basis?
16.2.2 The Reverse Ordering
16.2.3 Jordan Blocks
16.2.4 Jordan Blocks and Their Powers
16.3 General Matrix
16.3.1 Characteristic Polynomial: Eigenvalues and Their Multiplicity
16.3.2 Block and Its Invariant Subspace
16.3.3 Block and Its Jordan Form
16.3.4 Block and Its Characteristic Polynomial
16.4 Exercises: Hermitian Matrix and Its Eigenbasis
16.4.1 Nilpotent Hermitian Matrix
16.4.2 Hermitian Matrix and Its Orthonormal Eigenbasis
17 Jordan Decomposition of a Matrix
17.1 Greatest Common Divisor
17.1.1 Integer Division with Remainder
17.1.2 Congruence: Same Remainder
17.1.3 Common Divisor
17.1.4 The Euclidean Algorithm
17.1.5 The Extended Euclidean Algorithm
17.1.6 Confining the Coefficients
17.1.7 The Modular Extended Euclidean Algorithm
17.2 Modular Arithmetic
17.2.1 Coprime
17.2.2 Modular Multiplication
17.2.3 Modular Power
17.2.4 Modular Inverse
17.2.5 How to Find the Modular Inverse?
17.3 The Chinese Remainder Theorem
17.3.1 Modular Equation
17.3.2 How to Use the Coprime?
17.3.3 Modular System of Equations
17.3.4 Uniqueness
17.4 How to Use the Remainder?
17.4.1 Integer Decomposition
17.4.2 Binary Number
17.4.3 Horner's Algorithm
17.5 Polynomials and the Chinese Remainder Theorem
17.5.1 Characteristic Polynomial: Root and its Multiplicity
17.5.2 Multiplicity and Jordan Subspace
17.5.3 The Chinese Remainder Theorem with Polynomials
17.6 The Jordan Decomposition
17.6.1 Properties of Q
17.6.2 The Diagonal Part
17.6.3 The Nilpotent Part
17.6.4 The Jordan Decomposition
17.7 Example: Space of Polynomials
17.7.1 Polynomial and its Differentiation
17.7.2 The Jordan Basis
17.7.3 The Jordan Block
17.7.4 The Jordan Block and its Powers
17.8 Exercises: Numbers—Polynomials
17.8.1 Natural Numbers: Binary Form
18 Algebras and Their Derivations and Their Jordan Form
18.1 Eigenfunctions
18.1.1 Polynomials of Any Degree
18.1.2 Eigenfunction
18.1.3 The Leibniz (Product) Rule
18.1.4 Mathematical Induction
18.2 Algebra and Its Derivation
18.2.1 Leibniz Rule
18.2.2 Product (Multiplication)
18.2.3 Derivation
18.3 Product and Its Derivation
18.3.1 Two-Level (Virtual) Binary Tree
18.3.2 Multilevel Tree
18.3.3 Pascal's Triangle and the Binomial Formula
18.4 Product and Its Jordan Subspace
18.4.1 Two Members from Two Jordan Subspaces
18.4.2 Product and Its New Jordan Subspace
18.4.3 Example: Polynomials Times Exponents
18.5 Derivation on a Subalgebra
18.5.1 Restriction to a Subalgebra
18.6 More Examples
18.6.1 Smooth Functions
18.6.2 Finite Dimension
18.7 The Diagonal Part
18.7.1 Is It a Derivation?
18.8 Exercises: Derivation and Its Exponent
18.8.1 Exponent—Product Preserving
Part VII Linearization in Numerical Relativity
19 Einstein Equations and their Linearization
19.1 How to Discretize and Linearize?
19.1.1 How to Discretize in Time?
19.1.2 How to Differentiate the Metric?
19.1.3 The Flat Minkowski Metric
19.1.4 Riemann's Normal coordinates
19.1.5 Toward Gravity Waves
19.1.6 Where to Linearize?
19.2 The Christoffel Symbol
19.2.1 The Gradient Symbol and its Variation
19.2.2 The Inverse Matrix and its Variation
19.2.3 Einstein Summation Convention
19.2.4 The Christoffel Symbol and its Variation
19.3 Einstein Equations in Vacuum
19.3.1 The Riemann Tensor and its Variation
19.3.2 Vacuum and Curvature
19.3.3 The Ricci Tensor
19.3.4 Einstein Equations in Vacuum
19.3.5 Stable Time Marching
19.4 The Trace-Subtracted Form
19.4.1 The Stress Tensor
19.4.2 The Stress Tensor and its Variation
19.4.3 The Trace-Subtracted Form
19.5 Einstein Equations: General Form
19.5.1 The Ricci Scalar
19.5.2 The Einstein Tensor
19.5.3 Einstein Equations—General Form
19.6 How to Integrate?
19.6.1 Integration by Parts
19.6.2 Why to Linearize?
19.6.3 Back to the Trace-Subtracted Form
19.7 Exercises
References
Index