Numerical Linear Algebra with Applications

Chapter 5

Eigenvalues and Eigenvectors

Abstract

This chapter defines eigenvalues and their corresponding eigenvectors. It computes the eigenvalues and eigenvectors for a number of examples using polynomial root finding and Gaussian elimination with a homogeneous system. Some properties of eigenvalues are developed, including the fact that an n × n matrix is singular if and only if it has a zero eigenvalue, as well as the fact that the determinant of a matrix is the product of its eigenvalues. The concept of similar matrices is presented, and that similar matrices have the same eigenvalues. This discussion leads to the concept of matrix diagonalization. It is made clear that a matrix can be diagonalized only if it has n linearly independent ...