The thing to always keep in mind is the following definition of eigenvalues and eigenvectors. There are many ways to find them and in this chapter, I will present the standard way to do this. It is also the very worst way. This is a book on multivariable calculus, not one on linear algebra. This is why I have been focussed almost exclusively on ℝn. However, when one considers eigenvalues and eigenvectors, it is no longer possible to give a reasonable presentation without the use of the complex numbers. Thus, for the material in this section, it will be understood that the vectors are in ℂn meaning ordered lists of complex numbers. The matrices will also be understood to have entries in ℂ and all scalars will be understood to lie in ℂ rather than be restricted to be in ℝ.
Definition 19.1.1 Let A be an n×n matrix and let x ∈ ℂn,λ ∈ ℂ. Then x is an eigenvector for the eigenvalue λ if and only if the following two conditions hold.
Now here is an important observation which really is just a re statement of the above definition.
Theorem 19.1.2 Let A be an n×n matrix. The vector x is an eigenvector for the eigenvalue λ if and only if
Now with this fundamental definition, I will present the worst way of finding eigenvalues and eigenvectors. It is very important because everyone cherishes it and it is the standard way to do it in all undergraduate courses. Also, it gives an introduction to the important topic of determinants which will be presented in more detail later.