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 perform the QR decomposition. Next, the implicit single-shift Francis algorithm is discussed. The discussion relies on the implicit single-shift discussion in Chapter 18. The bisection method is a very clever technique that finds one eigenvalue, all the eigenvalues in a range, or all the eigenvalues of a symmetric matrix. It relies on the fact that the eigenvalues of the upper k × k submatrices of the tridiagonal matrix interlace. Also, there is a simple recurrence relation for the computation of the characteristic polynomials for those matrices. The number of sign changes as the n + 1 polynomials are evaluated at real number indicates the location of eigenvalues. Cuppen’s recursive divide-and-conquer method for computing all the eigenvalues and associated eigenvectors of a real symmetric matrix is outlined. No MATLAB implementation is provided; however, a C function using the MEX interface to the LAPACK routine is supplied. This allows testing of the algorithm’s speed and accuracy.