Chapter 8
Linear Systems and Eigenvalue Problems
Discretized differential equations and coupled physical systems often lead to linear equations, while oscillations and quantum states lead to eigenvalue problems. This chapter develops direct and iterative methods for linear systems, then turns to selected eigenvalues and the full spectrum of a symmetric matrix. It shows how matrix structure, conditioning, and the number of solutions needed guide the choice of method. Physical examples include spring chains, circuits, normal modes, and quantum energy levels.
In this chapter
- 8.1 Linear systems: introduction
- 8.2 Direct methods
- 8.3 Iterative methods
- 8.4 Practical notes for linear systems
- 8.5 Eigenvalue problems: introduction
- 8.6 Power methods
- 8.7 Jacobi method for symmetric matrices
- 8.8 QR method for symmetric matrices
- 8.9 Practical notes for eigenvalue problems
- 8.10 Computational exercises and projects
Companion resources
Codes
MATLAB, Python, C++, and Fortran programs for Gaussian elimination, SOR, the Thomas algorithm, matrix inversion, the power method, shifted inverse iteration, and Jacobi diagonalization.
View Chapter 8 codes →Exercises & Projects
Additional problems and physics projects for recognizing matrix structure, choosing a solver, checking accuracy, and interpreting solutions and modes.
View Chapter 8 exercises & projects →Resources
Benchmark matrices, method-selection guidance, numerical libraries for large problems, and selected references.
View Chapter 8 resources →Updates & Errata
Corrections, clarifications, and post-publication updates associated with Chapter 8.
View Chapter 8 updates & errata →