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CME 302 Numerical Linear Algebra
Class Notes 2025
Linear Algebra Bootcamp
Vector spaces
Dot Product and Vector Norms
Linear Transformations and Matrices
Matrix-Matrix Multiplications
Operator and Matrix Norms
The Sherman-Morrison-Woodbury Formula
The Determinant
The Trace of a Matrix
Orthogonal Matrices
Projections
Block Matrix Operations
The Four Fundamental Subspaces
Eigendecomposition
Normal matrices
Applications of Eigenvalues
Singular Value Decomposition
Eigenvalues and Singular Values
Summary of Matrix Decompositions
Solving Linear Systems
The LU Decomposition Algorithm
Existence and Uniqueness of LU Factorization
Floating-Point Numbers
LU Factorization with Row Pivoting
Cholesky Factorization
Least Squares Problems
Householder Reflections
Givens Rotations
Modified Gram-Schmidt
Summary
Existence and Uniqueness of QR Decomposition
Backward Stability of Householder and Givens QR
The QR Factorization and the Determinant
LU vs. QR Decomposition
The Method of Normal Equations
Solving Least-Squares using QR Factorization
SVD for Rank-Deficient Least-Squares
Summary of LS Solution Methods
Eigenvalue Computation
The Power Method
The Method of Deflation
The Orthogonal Iteration Algorithm
Power and Orthogonal Iteration Methods
Orthogonal Iteration and Eigenvalues
Computing Eigenvectors from the Schur Decomposition
QR Iteration
Reduction to Hessenberg Form
QR iteration for upper Hessenberg matrices
The Shifted QR Iteration Algorithm
Symmetric vs Unsymmetric QR Iteration
Iterative Methods for Eigenvalue Computation
Projection onto Krylov Subspaces
The Arnoldi Process
Convergence of the Arnoldi Process
The Lanczos Algorithm: Arnoldi for Symmetric Matrices
Convergence of the Lanczos Process
Classical Iterative Methods
Splitting Methods and Convergence Theory
The Jacobi Iteration Method
The Gauss-Seidel Iteration Method
Successive Over-Relaxation (SOR) Method
The Chebyshev Iteration Method
Krylov Subspace Methods
The Conjugate Gradient (CG) Method
The Generalized Minimal Residual Method (GMRES)
CG vs. GMRES
MINRES
Preconditioning
Preconditioned Conjugate Gradient
License for this book
Index