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