Eigenvalues and Eigenvectors
If the solution of a linear system is compared to the gift of fish, an analysis of the linear transformation represented by a matrix, in terms of its eigenvalues and eigenvectors, becomes comparable to instructions in the art of fishing. With a knowledge of eigenvalues and eigenvectors of a matrix, the complete behaviour of the transformation can be understood and explained with full insight and confidence. In this chapter, we carry out a foundational groundwork of the eigenvalue problem, the methods of which will be covered in detail in the following chapters.
Let us get back to our old theme of Chap. 2: a matrix represents a linear transformation, operates on vectors in its domain and maps them ...