Page 265 - Matrix Analysis & Applied Linear Algebra

260 Chapter 4 Vector Spaces

The invariant subspaces for a linear operator T are important because they
produce simplified coordinate matrix representations of T. To understand how
this occurs, suppose X is an invariant subspace under T, and let

B X = {x 1 , x 2 ,..., x r }
be a basis for X that is part of a basis

B = {x 1 , x 2 ,..., x r , y 1 , y 2 ,..., y q }
for the entire space V. To compute [T] B , recall from the definition of coordinate
matrices that

[T] B = [T(x 1 )] B ··· [T(x r )] B [T(y 1 )] B ··· [T(y q )] B . (4.9.1)

Because each T(x j ) is contained in X, only the first r vectors from B are
needed to represent each T(x j ), so, for j =1, 2,...,r,
 
α 1j
.
 . 
r  . 
 
 α rj
T(x j )= α ij x i and [T(x j )] B =   . (4.9.2)

 0 
i=1  . 
.
 . 
0
The space
Y = span {y 1 , y 2 ,..., y q } (4.9.3)
may not be an invariant subspace for T, so all the basis vectors in B may be
needed to represent the T(y j ) ’s. Consequently, for j =1, 2,...,q,
 
β 1j
 . 
.
 . 
r q  

 
T(y j )= β ij x i + γ ij y i and [T(y j )] B =  β rj  . (4.9.4)
 
i=1 i=1  γ 1j 
 . 
.
 . 
γ qj
Using (4.9.2) and (4.9.4) in (4.9.1) produces the block-triangular matrix
 
α 11 ··· α 1r β 11 ··· β 1q
 . . . . .
 . . . . . . . . . . . 
.
. 
 
 α r1 ··· α rr β r1 ··· 
[T] B =  β rq  . (4.9.5)
 0 ··· 0 γ 11 ··· 
 γ 1q 
 . . . . .
 . . . . . . . . . . . 
.
. 
0 ··· 0 γ q1 ··· γ qq