Introduction to Linear Algebra
Systems of Linear Equations
- Introduction
- Linear systems
- Vectors
- Linear combinations
- Matrices
- Planes in ℝ³
- Equation A x = b
- Sensitivity of solutions
- Linear independence
- Plane transformations
- Space transformations
- Linear transformations
- Affine maps
- Exercises
- Answers
Matrix Algebra
- Introduction
- Manipulation of matrices
- Partitioned matrices
- Block matrices
- Matrix operators
- Determinants
- Cofactors
- Cramer's rule
- Chiò's method
- Equivalent matrices
- Elimination: A = L U
- PLU factorization
- Reflection
- Givens rotation
- Special matrices
- Exercises
- Answers
Vector Spaces
- Introduction
- Motivation
- Vector spaces
- Bases
- Dimension
- Coordinate systems
- Linear transformations
- Change of basis
- Matrix transformations
- Compositions
- Isomorphisms
- Dual transformations
- Quotient spaces
- Rank
- Solving A x = b
- Exercises
- Answers
Eigenvalues, Eigenvectors
- Introduction
- Characteristic polynomials
- Companion matrix
- Algebraic and geometric multiplicities
- Minimal polynomials
- Eigenspaces
- Where are eigenvalues?
- Eigenvalues of A B and B A
- Generalized eigenvectors
- Similarity
- Diagonalizability
- Self-adjoint operators
- Exercises
- Answers
Euclidean Spaces
- Introduction
- Four spaces
- Euclidean space
- Bilinear transformations
- Norm and distance
- Matrix norms
- Condition numbers
- Dual norms
- Dual transformations
- Examples of transformations
- Orthogonality
- Gram--Schmidt Process
- Orthogonal matrices
- Self-adjoint matrices
- Unitary matrices
- Projection operators
- QR-decomposition
- Least Square Approximation
- Quadratic forms
- Exercises
- Answers
Canonical forms
- Introduction
- 2D decomposition
- 3D decomposition
- Projectors
- Direct-sum decompositions
- Cyclic decompositions
- Symmetric matrices
- Pseudoinverse
- URV-decomposition
- LU-decomposition
- QR-decomposition
- Cholesky decomposition
- Schur decomposition
- Positive matrices
- Roots
- Polar factorization
- Spectral decomposition
- CUR decomposition
- Exercises
- Answers
Applications
- Introduction
- Circles along curves
- TNB frames
- GPS problem
- Coriolis acceleration
- Poisson equation
- Graph theory
- Error correcting codes
- Electric circuits
- FSA
- Markov chains
- Cryptography
- Wave-length transfer matrix
- Computer graphics
- Linear Programming
- Hill's determinant
- Fibonacci matrices
- Discrete dynamic systems
- Discrete Fourier transform
- Fast Fourier transform
- Curve fitting
- Answers
Miscellany
- Introduction
- Circles along curves
- TNB frames
- Differential forms
- Calculus
- Vector representations
- Matrix representations
- Change of basis
- Orthonormal Diagonalization
- Generalized inverse
- Differential forms
Preliminaries
- Complex Number Operations
- Sets
- Polynomials
- Polynomials and Matrices
- Computer solves Systems of Linear Equations
- Location of Eigenvalues
- Power Method
- Iterative Method
- Similarity and Diagonalization
Glossary
Reference
This Book is licensed under Creative Commons Attribution-NonCommercial-NoDerivs 3.0 Unported License
Positive Definite Matrices
How to generate SPD matrix
B3 = (1/7)*{{3, 2, 6}, {-6, 3, 2}, {2, 6, -3}} C2 = (1/5)*{{3, 4}, {4, -3}} Q6 = KroneckerProduct[C2, B3] Q66 = Transpose[Q6] A6 = Q66 . DiagonalMatrix[{35, 70, 35, 105, 35, 70}] . Q6 A = A6*35 {{1901, 222, 372, 168, -504, -504}, {222, 2010, 150, -504, -420, 0}, {372, 150, 2557, -504, 0, -924}, {168, -504, -504, 1999, -72, 78}, {-504, -420, 0, -72, 1765, 150}, {-504, 0, -924, 78, 150, 2018}}