Page 270 - Matrix Analysis & Applied Linear Algebra

4.9 Invariant Subspaces 265

This extension might be preferred over that of Example 4.9.2 because the spaces
X = span {q 1 , q 2 } and Y = span {q 3 , q 4 } are both invariant under T, and
therefore it follows from (4.9.10) that Q −1 TQ is block diagonal. Indeed, it is
not difficult to verify that
1111 −1 −1 −1 −1 2 −1 0 0
     
 1222   0 −5 −16 −22   −1 2 −1
−1
Q 0 
1233   0 3 10 14   0 −1 2 −1 
TQ = 
1234 4 8 12 14 0 0 −1 1
12 00
 
 34 00 
=   .
00 56
 
00 78
Notice that the diagonal blocks must be the matrices of the restrictions in the
sense that

12 = T + and 56 = T + .
*
*
34 /X {q 1 ,q 2 } 78 /Y {q 3 ,q 4 }
Example 4.9.4
2
Problem: Find all subspaces of that are invariant under

0 1
A = .
−2 3
Solution: The trivial subspace {0} is the only zero-dimensional invariant sub-
2
space, and the entire space is the only two-dimensional invariant subspace.
The real problem is to find all one-dimensional invariant subspaces. If M is a
one-dimensional subspace spanned by x = 0 such that A(M) ⊆M, then
Ax ∈M =⇒ there is a scalar λ such that Ax = λx =⇒ (A − λI) x = 0.

In other words, M⊆ N (A − λI) . Since dim M =1, it must be the case that
N (A − λI) = {0}, and consequently λ must be a scalar such that (A − λI)is
a singular matrix. Row operations produce

−λ 1 −2 3 − λ −2 3 − λ
A − λI = −→ −→ 2 ,
−2 3 − λ −λ 1 0 1+(λ − 3λ)/2
2
and it is clear that (A − λI) is singular if and only if 1 + (λ −3λ)/2 = 0 —i.e.,
if and only if λ is a root of
2
λ − 3λ +2 = 0.