Page 237 - 35Linear Algebra

12.3 Eigenspaces 237

is also an eigenvector of L with eigenvalue 1. In fact, any linear combination

   
−1 1
0
r  1  + s  
0 1
of these two eigenvectors will be another eigenvector with the same eigen-
value.
More generally, let {v 1 , v 2 , . . .} be eigenvectors of some linear transforma-
tion L with the same eigenvalue λ. A linear combination of the v i is given
1
2
2
1
by c v 1 + c v 2 + · · · for some constants c , c , . . .. Then
1
1
2
2
L(c v 1 + c v 2 + · · · ) = c Lv 1 + c Lv 2 + · · · by linearity of L
1 2
= c λv 1 + c λv 2 + · · · since Lv i = λv i
2
1
= λ(c v 1 + c v 2 + · · · ).
So every linear combination of the v i is an eigenvector of L with the same
eigenvalue λ. In simple terms, any sum of eigenvectors is again an eigenvector
if they share the same eigenvalue.
The space of all vectors with eigenvalue λ is called an eigenspace. It
is, in fact, a vector space contained within the larger vector space V . It
contains 0 V , since L0 V = 0 V = λ0 V , and is closed under addition and scalar
multiplication by the above calculation. All other vector space properties are
inherited from the fact that V itself is a vector space. In other words, the
subspace theorem (9.1.1, chapter 9) ensures that V λ := {v ∈ V |Lv = 0} is a
subspace of V .

Eigenspaces

Reading homework: problem 3

You can now attempt the second sample midterm.

237