Numerical Linear Algebra with Applications

Chapter 19

The Symmetric Eigenvalue Problem

Abstract

The chapter presents five algorithms for the computation of eigenvalues and, in most cases, their associated eigenvectors of a symmetric matrix. After proving the spectral theorem and reviewing properties of a symmetric matrix, the Jacobi algorithm is presented in detail, including proving convergence. Following that, the algorithm that uses Householder reflections to orthogonally transform a symmetric matrix to a symmetric tridiagonal matrix is discussed. The remaining algorithms all require this initial step. The Wilkinson shift is introduced and the single-shift symmetric QR iteration is presented that transforms the symmetric tridiagonal matrix to a diagonal matrix. Givens rotations ...