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Summary Numerical Linear Algebra — Complete Module Revision Guide

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13 pages
2025/2026 Written for academic year
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The University Of Edinburgh (ED)
Course
Numerical Linear Algebra
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Numerical Linear Algebra — an applied mathematics/numerical analysis module covering the theory and algorithms behind solving linear systems, least squares problems, and eigenvalue problems numerically. Written as a complete, restructured revision guide (not raw lecture notes) — organised by topic for efficient exam preparation, covering the full syllabus in 13 pages with full derivations, proofs, and worked examples throughout. Index of topics covered: 1. Foundations: floating point arithmetic, algorithm cost, vector and matrix norms 2. Direct methods: Gaussian elimination, LU factorisation, partial pivoting (GEPP), condition numbers 3. QR factorisation (Gram-Schmidt, Modified Gram-Schmidt, Householder reflections) 4. Iterative methods for linear systems (Jacobi, Gauss-Seidel) 5. Least squares (normal equations, LSQ-QR, LSQ-SVD) 6. The singular value decomposition (SVD) 7. Eigenvalue problems (power iteration, shifted inverse iteration, orthogonal iteration) This guide includes full mathematical proofs (not just definitions) for the key theorems of the course — including the correctness of PA=LU factorisation, the backward stability bounds for GEPP and Householder QR, the normal equations theorem, the condition number formula κ₂(A)=σmax/σmin, and the convergence proof for power iteration. Suitable for students studying numerical analysis, numerical linear algebra, or applied/computational mathematics modules with similar content.

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Numerical Linear Algebra
Complete Revision & Study Guide

A topic-by-topic guide covering Gaussian elimination, QR factorisation, iterative methods,
least squares, the singular value decomposition, and eigenvalue problems — with full
derivations, proofs, and worked examples throughout.



Original study notes — independently written summary and explanation

, Contents

• 1. Foundations: Floating Point, Cost, and Norms
• 2. Direct Methods: Gaussian Elimination and LU Factorisation
• 3. QR Factorisation
• 4. Iterative Methods for Linear Systems
• 5. Least Squares
• 6. The Singular Value Decomposition
• 7. Eigenvalue Problems

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Browse related subjects

Subjects

  • numerical linear algebra
  • floating point arithmetic
  • machine epsilon
  • matrix norms
  • condition number
  • gaussian elimination
  • lu factorisation
  • gepp
  • qr factorisation
  • backward stability
  • partial pivoting
  • jacobi

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