Page 268 - Matrix Analysis & Applied Linear Algebra

4.9 Invariant Subspaces 263

Triangular and Diagonal Block Forms
When T is an n × n matrix, the following two statements are true.

• Q is a nonsingular matrix such that

Q −1 TQ = A r×r B r×q (4.9.9)
0 C q×q

if and only if the first r columns in Q span an invariant subspace
under T.

• Q is a nonsingular matrix such that
0 0
 
A r 1 ×r 1 ···
0 B r 2 ×r 2 ··· 0 

Q −1 TQ =  . . . .  (4.9.10)
. . . .
 . . . . 
0 0 ··· C r k ×r k

if and only if Q = Q 1 Q 2 ··· Q k in which Q i is n × r i , and

the columns of each Q i span an invariant subspace under T.
Proof. We know from Example 4.8.3 that if B = {q 1 , q 2 ,..., q n } is a basis for
n
, and if Q = q 1 q 2 ··· q n is the matrix containing the vectors from B

as its columns, then [T] B = Q −1 TQ. Statements (4.9.9) and (4.9.10) are now
direct consequences of statements (4.9.7) and (4.9.8), respectively.
Example 4.9.2
Problem: For
−1 −1 −1 −1 2 −1
     
 0 −5 −16 −22   −1  and  2 
 ,
 ,
 ,
0 3 10 14 q 1 =  0 q 2 =  −1
T = 
4 8 12 14 0 0
verify that X = span {q 1 , q 2 } is an invariant subspace under T, and then find
a nonsingular matrix Q such that Q −1 TQ has the block-triangular form
 
∗ ∗ ∗ ∗
 ∗ ∗ ∗ ∗ 
−1
Q TQ =   .
0 0 ∗ ∗
 
0 0 ∗ ∗