Page 267 - Matrix Analysis & Applied Linear Algebra

262 Chapter 4 Vector Spaces

then the r -dimensional subspace U = span {u 1 , u 2 ,..., u r } spanned by the
first r vectors in B must be an invariant subspace under T. Furthermore, if
the matrix representation of T has a block-diagonal form

0
A r×r
[T] B =
0 C q×q
relative to B, then both
U = span {u 1 , u 2 ,..., u r } and W = span {w 1 , w 2 ,..., w q }
must be invariant subspaces for T. The details are left as exercises.
The general statement concerning invariant subspaces and coordinate ma-
trix representations is given below.

Invariant Subspaces and Matrix Representations
Let T be a linear operator on an n-dimensional space V, and let
X, Y,..., Z be subspaces of V with respective dimensions r 1 ,r 2 ,...,r k
and bases B X , B Y ,..., B Z . Furthermore, suppose that i i = n and
r
B = B X ∪B Y ∪· · · ∪B Z is a basis for V.

• The subspace X is an invariant subspace under T if and only if
[T] B has the block-triangular form

B * +
A r 1 ×r 1
[T] B = , in which case A = T . (4.9.7)
0 C /X B X
• The subspaces X, Y,..., Z are all invariant under T if and only if
[T] B has the block-diagonal form
0 0
 
A r 1 ×r 1 ···
0 ··· 0
 B r 2 ×r 2 
[T] B =  . . . .  , (4.9.8)
 . . . . . . . . 
0 0 ··· C r k ×r k
in which case

* + * + * +
A = T , B = T , . . . , C = T .
/X B x /Y B y /Z B z

An important corollary concerns the special case in which the linear operator
T is in fact an n × n matrix and T(v)= Tv is a matrix–vector multiplication.