Scope: Saad → Numerical Linear Algebra → Krylov Methods → Numerical PDE / FEM

Complete roadmap

Study order

• 1. Scalars, Vectors, Matrices and Maps
• 1. Vector Spaces and Subspaces
• 1. Linear Maps
• 1. Rank and Fundamental Subspaces
• 1. Quotient and Isomorphism Viewpoint
• 1. Inner Product Geometry
• 1. Orthogonal Decomposition
• 1. Eigenvalues and Eigenvectors
• 1. Diagonalization
• 1. Special Matrix Classes
• 1. Spectral Theorem
• 1. Singular Value Decomposition
• 1. Matrix and Operator Norms
• 1. Conditioning
• 1. Matrix Factorizations
• 1. Schur Decomposition
• 1. Jordan and Canonical Structure
• 1. Polynomial Viewpoint
• 1. Krylov Bridge
• 1. Bridge to Numerical PDE

Dependency

11: SVD existence / rank → 12: Norms → 11: Low-rank approximation / Eckart–Young → 13: Conditioning

Maps → Subspaces → Quotients → Inner products → Spectral structure → SVD / Norms → Conditioning → Factorizations → Matrix polynomials → Krylov / PDE

Cross-reference

• Folland Seminar
• Saad
• Numerical PDE / FEM