Page 204 - Linear Algebra Done Right

Chapter 9. Operators on Real Vector Spaces
194
Eigenvalues of Square Matrices
We have defined eigenvalues of operators; now we need to extend
that notion to square matrices. Suppose A is an n-by-n matrix with
entries in F. A number λ ∈ F is called an eigenvalue of A if there
exists a nonzero n-by-1 matrix x such that
Ax = λx.

For example, 3 is an eigenvalue of 78 because
15

7 8 2 6 2
= = 3 .
1 5 −1 −3 −1

As another example, you should verify that the matrix 0 −1 has no
10
eigenvalues if we are thinking of F as the real numbers (by definition,
an eigenvalue must be in F) and has eigenvalues i and −i if we are
thinking of F as the complex numbers.
We now have two notions of eigenvalue—one for operators and one
for square matrices. As you might expect, these two notions are closely
connected, as we now show.

9.1 Proposition: Suppose T ∈L(V) and A is the matrix of T with
respect to some basis of V. Then the eigenvalues of T are the same as
the eigenvalues of A.

Proof: Let (v 1 ,...,v n ) be the basis of V with respect to which T
has matrix A. Let λ ∈ F. We need to show that λ is an eigenvalue of T
if and only if λ is an eigenvalue of A.
First suppose λ is an eigenvalue of T. Let v ∈ V be a nonzero vector
such that Tv = λv. We can write

9.2 v = a 1 v 1 +· · ·+ a n v n ,

where a 1 ,...,a n ∈ F. Let x be the matrix of the vector v with respect
to the basis (v 1 ,...,v n ). Recall from Chapter 3 that this means

 
a 1
 . 
9.3 x =  . .  .
 
a n