Page 266 - Matrix Analysis & Applied Linear Algebra

4.9 Invariant Subspaces 261

r

The equations T(x j )= α ij x i in (4.9.2) mean that
i=1
   
α 1j α 11 α 12 ··· α 1r
 α 21 α 22
* + * +
 α 2j  ··· α 2r 
T (x j ) =  .  , so T =  . . . .  ,
/X  .   . . . . . 
B X . /X B X . . .
α rj α r1 α r2 ··· α rr
and thus the matrix in (4.9.5) can be written as
* +
T B r×q
[T] B = /X B X . (4.9.6)
0 C q×q
In other words, (4.9.6) says that the matrix representation for T can be made
to be block triangular whenever a basis for an invariant subspace is available.
The more invariant subspaces we can find, the more tools we have to con-
struct simplified matrix representations. For example, if the space Y in (4.9.3)
is also an invariant subspace for T, then T(y j ) ∈Y for each j =1, 2,...,q,
and only the y i ’s are needed to represent T(y j ) in (4.9.4). Consequently, the
β ij ’s are all zero, and [T] B has the block-diagonal form

* +
 
T 0

0
A r×r /X B x
[T] B = =  * +  .
0 C q×q 0 T /Y B y
This notion easily generalizes in the sense that if B = B X ∪B Y ∪· · ·∪B Z is a basis
for V, where B X , B Y ,..., B Z are bases for invariant subspaces under T that
have dimensions r 1 ,r 2 ,...,r k , respectively, then [T] B has the block-diagonal
form
0 ··· 0
 
A r 1 ×r 1
0 ··· 0
 B r 2 ×r 2 
[T] B =  . . . .  ,
. . .
 . . . . . 
0 0 ··· C r k ×r k
where
* + * + * +
A = T /X B x , B = T /Y B y , . .., C = T /Z B z .
The situations discussed above are also reversible in the sense that if the
matrix representation of T has a block-triangular form

A r×r B r×q
[T] B =
0 C q×q
relative to some basis

B = {u 1 , u 2 ,..., u r , w 1 , w 2 ,..., w q },