---
title: Eigenvalue Problems
description: Eigenvalue problems involve finding vectors whose direction remains unchanged under a given linear transformation. Learners will understand how to compute eigenvalues and eigenvectors to analyze dynamical systems and structural stability.
category: mathematics
subcategory: numerical-linear-algebra
difficulty: beginner, intermediate, advanced
url: /subject/eigenvalue-problems
---

# Eigenvalue Problems

Eigenvalue problems involve finding vectors whose direction remains unchanged under a given linear transformation. Learners will understand how to compute eigenvalues and eigenvectors to analyze dynamical systems and structural stability.

## Available Resources

3 Books • 2 Websites

## Websites

### 1. LAPACK Users' Guide (Third Edition)

**Author:** E. Anderson, Z. Bai, J. Demmel, J. Dongarra, et al.

The official LAPACK manual, free in full on netlib. Documents the driver and computational routines for symmetric, nonsymmetric, and generalized eigenproblems and the SVD, with the accuracy and error-bound analysis attached to each routine.

**Difficulty:** Intermediate | **Language:** English | **Price:** Free

**Link:** https://www.netlib.org/lapack/lug/

**Tags:** lapack, eigenvalue-problems, numerical-linear-algebra, singular-value-decomposition, scientific-computing

### 2. Wolfram MathWorld

**Author:** Eric W. Weisstein

MathWorld is an online mathematics encyclopedia from Wolfram Research offering detailed, browsable articles on topics across the math spectrum, including algebra, geometry, calculus, and number theory. Each entry includes definitions, theorems, formulas, diagrams, worked examples, and links to further reading.

**Difficulty:** Intermediate | **Language:** English | **Price:** Free

**Link:** https://mathworld.wolfram.com

**Tags:** mathematics-reference, encyclopedia, abstract-algebra, number-theory, geometry

## Books

### 1. Templates for the Solution of Algebraic Eigenvalue Problems

**Author:** Zhaojun Bai, James Demmel, Jack Dongarra, Axel Ruhe, Henk van der Vorst (eds.)

Answers the question the textbooks dodge: given MY matrix, which eigensolver do I actually call? The decision-tree structure plus per-algorithm templates and software pointers makes it the practical bridge from theory to LAPACK/ARPACK. Full text, free on netlib, of the SIAM practical guide edited by Bai, Demmel, Dongarra, Ruhe, and van der Vorst. An informal decision tree routes Hermitian, generalized, non-Hermitian, and nonlinear eigenproblems to specific algorithms and software.

**Difficulty:** Advanced | **Language:** English | **Price:** Free

**Link:** https://www.netlib.org/utk/people/JackDongarra/etemplates/book.html

**Tags:** eigenvalue-algorithms, numerical-linear-algebra, sparse-matrices, lapack, arpack

### 2. The Symmetric Eigenvalue Problem (SIAM Classics)

**Author:** Beresford N. Parlett

The symmetric case is where most real applications live (structural stability, vibration modes, covariance/PCA) and this is the one book that treats it exhaustively rather than as a chapter. A monograph devoted entirely to computing eigenvalues of real symmetric matrices, covering Rayleigh quotient iteration, Sturm sequences and bisection, tridiagonalization, the QL/QR algorithms, and Lanczos convergence theory. SIAM Classics reprint of the 1980 Prentice-Hall original.

**Difficulty:** Advanced | **Language:** English | **Price:** Paid

**Link:** https://www.amazon.com/dp/0898714028?tag=edmonddante07-20

**Tags:** symmetric-eigenvalue-problem, numerical-linear-algebra, lanczos-algorithm, qr-algorithm, rayleigh-quotient-iteration

### 3. Numerical Methods for Large Eigenvalue Problems (2nd Edition)

**Author:** Yousef Saad

Saad's SIAM Classics monograph on computing eigenvalues of large sparse matrices: perturbation theory and conditioning, power and subspace iteration, Lanczos, Arnoldi, Davidson and Jacobi-Davidson, plus preconditioning. The best free text on the large/sparse computational eigenproblem, which is what 'numerical' linear algebra means at scale. Complete second-edition PDF hosted by the author at Minnesota.

**Difficulty:** Advanced | **Language:** English | **Price:** Free

**Link:** https://www-users.cse.umn.edu/~saad/eig_book_2ndEd.pdf

**Tags:** sparse-matrices, krylov-subspace-methods, arnoldi-method, jacobi-davidson, numerical-linear-algebra

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