Chapter 19

Eigenvalues and Eigenvectors19.1 Definition of Eigenvalues

The 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. This is a book on multi-variable calculus, not oneon 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 reason-able presentation without the use of the complex numbers. Thus, for the material in thissection, it will be understood that the vectors are in Cn meaning ordered lists of complexnumbers. The matrices will also be understood to have entries in C and all scalars will beunderstood 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 aneigenvector for 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 although0 can be an eigenvalue.

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

Theorem 19.1.2 Let A be an n× n matrix. The vector x is an eigenvector for theeigenvalue λ if and only if (A−λ I)−1 does not exist.

Proof: If (A−λ I)−1 does not exist, then by Theorem 18.5.12 the columns of A −λ I are 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 becauseits 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 eigen-values and eigenvectors. It is very important because everyone cherishes it and it is thestandard way to do it in all undergraduate courses. Also, it gives an introduction to theimportant topic of determinants which will be presented in more detail later.

19.2 An Introduction to Determinants19.2.1 Cofactors and 2×2 DeterminantsLet A be an n× n matrix. The determinant of A, denoted as det(A) is a number. If thematrix is a 2×2 matrix, this number is very easy to find.

Definition 19.2.1 Let A =

(a bc d

). Then det(A)≡ ad−cb. The determinant is

also often denoted by enclosing the matrix with two vertical lines. Thus

det(

a bc d

)=

∣∣∣∣ a bc d

∣∣∣∣ .409