Page 264 - Matrix Analysis & Applied Linear Algebra

4.9 Invariant Subspaces 259
4.9 INVARIANT SUBSPACES

For a linear operator T on a vector space V, and for X⊆ V,
T(X)= {T(x) | x ∈X}
is the set of all possible images of vectors from X under the transformation T.
Notice that T(V)= R (T). When X is a subspace of V, it follows that T(X)
is also a subspace of V, but T(X) is usually not related to X. However, in
some special cases it can happen that T(X) ⊆X, and such subspaces are the
focus of this section.

Invariant Subspaces
• For a linear operator T on V, a subspace X⊆ V is said to be an
invariant subspace under T whenever T(X) ⊆X.
• In such a situation, T can be considered as a linear operator on X
by forgetting about everything else in V and restricting T to act
only on vectors from X. Hereafter, this restricted operator will
be denoted by T .
/X

Example 4.9.1
Problem: For
     
4 4 4 2 −1
A =  −2 −2 −5  , x 1 =  −1  , and x 2 =  2  ,
1 2 5 0 −1
show that the subspace X spanned by B = {x 1 , x 2 } is an invariant subspace
under A. Then describe the restriction A and determine the coordinate
/X
matrix of A relative to B.
/X
Solution: Observe that Ax 1 =2x 1 ∈X and Ax 2 = x 1 +2x 2 ∈X, so the
image of any x = αx 1 + βx 2 ∈X is back in X because

Ax = A(αx 1 +βx 2 )= αAx 1 +βAx 2 =2αx 1 +β(x 1 +2x 2 )=(2α+β)x 1 +2βx 2 .
This equation completely describes the action of A restricted to X, so
A (x)=(2α + β)x 1 +2βx 2 for each x = αx 1 + βx 2 ∈X.
/X
Since A (x 1 )=2x 1 and A (x 2 )= x 1 +2x 2 , we have
/X /X

21
* + * + * +
A = A (x 1 ) A (x 2 ) = .

/X /X 02
/X B
B B