Matrix Computations: Eigenvalues and Eigenvectors

This lecture explores the complexity of matrix computations, focusing on computing eigenvalues and eigenvectors of diagonalizable and nonsymmetric matrices. It discusses algorithms for reducing matrices to Hessenberg and real Schur forms, analyzing the errors in the computations. The concept of nearby matrices and the behavior of eigenvalues under perturbations are also examined. Additionally, the lecture delves into the diagonalizability of matrices, forward and backward errors in computing eigenvectors, and the significance of numerical stability. Various algorithms for computing eigenvalues are presented, emphasizing practical considerations and theoretical complexities.