Page 200 - DMTH502_LINEAR_ALGEBRA

Linear Algebra

Notes V a unique k-tuple ( ,...,  ) of vectors  in W (by  =  + ... +  ) in such a way that we can carry
1 k i i 1 k
out the linear operations in V by working in the individual subspaces W . The fact that each W is
i i
invariant under T enables us to view the action of T as the independent action of the operators T
i
on the subspaces W . Our purpose is to study T by finding invariant direct-sum decompositions
i
in which the T are operators of an elementary nature.
i
Before looking at an example, let us note the matrix analogue of this situation. Suppose we select
an ordered basis  for each W , and let it be the ordered basis for V consisting of the union of the
i i
 arranged in the order  ,...,  , so that  is a basis for V. From our discussion concerning the
i 1 k
matrix analogue for a single invariant subspace, it is easy to see that if A = [T] and A = [T ] , then
 i i 
A has the block form
 A 1 0  0 
 0 A  0 
A   2 
     ...(1)
  
 0 0 A k
In (1), A is a d × d matrix (d = dim W ), and the 0’s are symbols for rectangular blocks of scalar
i i i i i
0’s of various sizes. It also seems appropriate to describe (1) by saying that A is the direct sum of
the matrices A ,..., A .
1 k
Most often, we shall describe the subspace W by means of the associated projections E (Theorem
i i
1 of unit 16). Therefore, we need to be able to phrase the invariance of the subspaces W in terms
i
of the E .
i
17.2 Some Theorems

Theorem 1: Let T be a linear operator on the space V, and W ,..., W and E ,..., E be as in Theorem
1 k 1 k
1 of unit 16. Then a necessary and sufficient condition that each subspace W be invariant under
i
T is that T commutes with each of the projections E , i.e.,
i
TE = E T, i = 1,..., k
i i
Proof: Suppose T commutes with each E . Let  be in W . Then E  = , and
i j j
T = T(E )
j
= E (T)
j
which shows that T is in the range of E , i.e., that W is invariant under T.
j j
Assume now that each W is invariant under T. We shall show that TE = E T. Let  be any vector
i j j
in V. Then
 = E  + ... + E 
1 k
T = TE  + ... + TE 
1 k
Since E  is in W , which is invariant under T, we must have T(E ) = E  for some vector  . Then
i i i i i i
E TE  = E E
j i j i i
 0, if i  j
= 
 E  j , if i  j
j
Thus
E T = E TE  + ... + E TE 
j j 1 j k
= E 
j j

194 LOVELY PROFESSIONAL UNIVERSITY