Page 271 - Matrix Analysis & Applied Linear Algebra

266 Chapter 4 Vector Spaces

Thus λ = 1 and λ =2, and straightforward computation yields the two one-
dimensional invariant subspaces

1 1
M 1 = N (A − I)= span and M 2 = N (A − 2I)= span .
1 2
1
1
( )
2
In passing, notice that B = , is a basis for , and
1 2

10 11
−1
[A] B = Q AQ = , where Q = .
02 12
In general, scalars λ for which (A − λI) is singular are called the eigenvalues
of A, and the nonzero vectors in N (A − λI) are known as the associated
eigenvectors for A. As this example indicates, eigenvalues and eigenvectors
are of fundamental importance in identifying invariant subspaces and reducing
matrices by means of similarity transformations. Eigenvalues and eigenvectors
are discussed at length in Chapter 7.

Exercises for section 4.9

4.9.1. Let T be an arbitrary linear operator on a vector space V.
(a) Is the trivial subspace {0} invariant under T?
(b) Is the entire space V invariant under T?

4.9.2. Describe all of the subspaces that are invariant under the identity oper-
ator I on a space V.

4
4.9.3. Let T be the linear operator on defined by
T(x 1 ,x 2 ,x 3 ,x 4 )=(x 1 + x 2 +2x 3 − x 4 ,x 2 + x 4 , 2x 3 − x 4 ,x 3 + x 4 ),

and let X = span {e 1 , e 2 } be the subspace that is spanned by the first
4
two unit vectors in .
(a) Explain why X is invariant under T.
" #
(b) Determine T .
/X {e 1 ,e 2 }
(c) Describe the structure of [T] B , where B is any basis obtained
from an extension of {e 1 , e 2 } .