Linear Algebra with Applications, 10th Edition

7.6 The Eigenvalue Problem

In this section, we are concerned with numerical methods for computing the eigenvalues and eigenvectors of an $n\times n$ matrix A. The first method we study is called the power method. The power method is an iterative method for finding the dominant eigenvalue of a matrix and a corresponding eigenvector. By the dominant eigenvalue, we mean an eigenvalue ${\lambda }_1$ satisfying $\left|{\lambda }_1\right|>\left|{\lambda }_i\right|$ for $i=2,\mathrm{\dots },n$. If the eigenvalues of A satisfy

$$\left|{\lambda }_1\right|>\left|{\lambda }_2\right|>\cdot \cdot \cdot >\left|{\lambda }_n\right|$$

then the power method can be used to compute the eigenvalues one at a time. The second method, the QR algorithm, is an iterative method involving orthogonal similarity transformations. It has many advantages over the power method. It will converge whether or not A has a dominant eigenvalue, ...