O'Reilly logo

Numerical Linear Algebra with Applications by William Ford

Stay ahead with the world's most comprehensive technology and business learning platform.

With Safari, you learn the way you learn best. Get unlimited access to videos, live online training, learning paths, books, tutorials, and more.

Start Free Trial

No credit card required

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 ...

With Safari, you learn the way you learn best. Get unlimited access to videos, live online training, learning paths, books, interactive tutorials, and more.

Start Free Trial

No credit card required