Chapter 10

Eigenvalues and Eigenvectors

10.1 Definition of EigenvaluesThe thing to always keep in mind is the following definition of eigenvalues and eigenvec-tors. There are many ways to find them and in this chapter, I will present the standard wayto do this. It is also the very worst way.

Definition 10.1.1 Let A be an n×n matrix and let x∈Cn,λ ∈C. Then x is an eigenvectorfor the eigenvalue λ if and only if the following two conditions hold.

1. Ax= λx

2. x ̸= 0. This is very important. By definition 0 is NEVER an eigenvector althoughit can be an eigenvalue.

Now here is an important observation which really is just a re statement of the abovedefinition.

Theorem 10.1.2 Let A be an n×n matrix. The vector x is an eigenvector for the eigenvalueλ if and only if (A−λ I)−1 does not exist.

Proof: If (A−λ I)−1 does not exist, then by Theorem 9.1.17 the columns of A− λ Iare not independent because its rank is less than n. Thus there exists x ̸= 0 such that(A−λ I)x= 0 and so λ is an eigenvalue and x is an eigenvector which goes with λ .Conversely, if (A−λ I)x= 0, and x ̸= 0, then the rank of (A−λ I) has no inverse be-cause its rank is less than n. Indeed, some column is a linear combination of the others.■

Now with this fundamental definition, I will present the worst way of finding eigenval-ues and eigenvectors. It is very important because everyone cherishes it. Also, it gives anintroduction to the important topic of determinants which will be presented in more detaillater.

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