Page 272 - Matrix Analysis & Applied Linear Algebra

4.9 Invariant Subspaces 267

4.9.4. Let T and Q be the matrices
−2 −1 −5 −2 1 0 0 −1
   
 −9 0 −8 −2   1 1 3 −4 
and  .
2 3 11 5  Q =  −2 0 1 0
T = 
3 −5 −13 −7 3 −1 −4 3
4
(a) Explain why the columns of Q are a basis for .
(b) Verify that X = span {Q ∗1 , Q ∗2 } and Y = span {Q ∗3 , Q ∗4 }
are each invariant subspaces under T.
(c) Describe the structure of Q −1 TQ without doing any compu-
tation.
(d) Now compute the product Q −1 TQ to determine
* + * +
T and T .
/X {Q ∗1 ,Q ∗2 } /Y {Q ∗3 ,Q ∗4 }

4.9.5. Let T be a linear operator on a space V, and suppose that
B = {u 1 ,..., u r , w 1 ,..., w q }
is a basis for V such that [T] B has the block-diagonal form

0
A r×r
[T] B = .
0 C q×q
Explain why U = span {u 1 ,..., u r } and W = span {w 1 ,..., w q } must
each be invariant subspaces under T.
4.9.6. If T n×n and P n×n are matrices such that

0
A r×r
−1
P TP = ,
0 C q×q
explain why
U = span {P ∗1 ,..., P ∗r } and W = span {P ∗r+1 ,..., P ∗n }
are each invariant subspaces under T.

4.9.7. If A is an n × n matrix and λ is a scalar such that (A − λI)is
singular (i.e., λ is an eigenvalue), explain why the associated space of
eigenvectors N (A − λI) is an invariant subspace under A.

−9 4
4.9.8. Consider the matrix A = .
−24 11
(a) Determine the eigenvalues of A.
2
(b) Identify all subspaces of that are invariant under A.
(c) Find a nonsingular matrix Q such that Q −1 AQ is a diagonal
matrix.