To illustrate why Parlett’s text is so valuable, consider the problem of computing eigenvectors of nearly multiple eigenvalues. Standard textbooks say “the eigenvectors become ill-conditioned.” Parlett says:
“When eigenvalues cluster, the eigenvectors are not individually meaningful; only their invariant subspace is well-determined. Any rotation of an orthonormal basis for that subspace is also a valid eigenbasis.”
He then introduces the canonical angles between subspaces (the sin(Θ) metric) to measure how close two invariant subspaces are. This geometric viewpoint directly informs algorithms: if you only need the subspace (e.g., for PCA), you can stop early without computing individual eigenvectors. parlett the symmetric eigenvalue problem pdf
No other book on symmetric eigenvalues gives such a clear geometric and numerical treatment of subspaces.
Supplement with lecture notes or Trefethen & Bau (for computational intuition) before tackling Parlett. To illustrate why Parlett’s text is so valuable,
Thus, Parlett is best paired with a modern implementation guide (e.g., Golub & Van Loan’s Matrix Computations or Demmel’s Applied Numerical Linear Algebra).
Chapters 1-3 lay the foundation. Parlett avoids simple matrix multiplication; instead, he focuses on invariant subspaces rather than individual eigenvectors. Key concepts include: He then introduces the canonical angles between subspaces
Chapters 4-7 cover the “direct” methods that transform ( A ) into tridiagonal form using orthogonal matrices (Householder or Givens rotations). Topics include:
Parlett’s treatment of the ( QR ) algorithm is particularly celebrated: he explains how Wilkinson’s shifts achieve cubic convergence without mysticism.
Title: The Symmetric Eigenvalue Problem
Author: Beresford N. Parlett
Series: Classics in Applied Mathematics (SIAM)
Original Publication: 1980 (SIAM edition 1998)
This book is a definitive, rigorous, and practical treatment of numerical methods for computing eigenvalues and eigenvectors of symmetric (and Hermitian) matrices. It is widely considered the canonical reference in the field, bridging pure linear algebra, numerical analysis, and software implementation.