Eigenvalues and Eigenvectors

Abstract

This chapter begins with the basic theory of eigenvalues and eigenvectors of matrices. Essential concepts such as characteristic polynomials, the Fundamental Theorem of Algebra, the Gerschgorin circle theorem, invariant subspaces, change of basis, spectral radius and the distance between subspaces are developed. Hermitian matrices are analyzed through the spectral theorem, and a perturbation analysis of their eigenvalue problem is performed. This chapter presents and examines algorithms for finding eigenvalues of Hermitian tridiagonal matrices, such as bisection, the power method, QL, QR, implicit QR, divide and conquer and dqds. Reduction of general Hermitian matrices to tridigonal form, and the Lanczos process are also discussed. Next, the eigenvalue problem for general matrices is examined. Theory for the Schur decomposition and the Jordan form are presented. Perturbation theory and conditions numbers lead to a posteriori estimates for general eigenvalue problems. Numerical methods for upper Hessenberg matricces are discussed, followed by general techniques for orthogonal similarity transformation to upper Hessenberg form. Then the chapter turns to the singular value decomposition, with theory discussing its existence, pseudo-inverses and the minimax theorem. Methods for reducing general matrices to bidiagonal form, and techniques for finding singular value decompositions of bidiagonal matrices follow next. The chapter ends with discussions of linear recurrences and functions of matrices.