Page 226 - DMTH502_LINEAR_ALGEBRA

Linear Algebra

Notes The cyclic subspace Z( ; T) is the subspace spanned by the vectors t , as i runs over those indices
j j
for which d j. The T-annihilator of is
i j
p j (x c i ). ...(21)
i d j
We have

V ( Z ; ) ... Z ( ; )
T
T
1 r
because each t j belongs to one and only one of the subspaces Z( ; T); ...., Z( ; T) and
1
r
= ( , ...., ) is a basis for V. By (21) p divides p .
1 k j + 1 j
Self Assessment
3
1. Let T be the linear operator on R which is represented in the standard ordered basis by
1 3 3
3 1 3
3 3 5

Find the characteristic polynomial for T. What is the minimal polynomial?
2. Show that if T is a diagonalizable linear operator then every T-invariant subspace has a
complementary T-invariant subspace.

20.4 Summary

 In this unit the theorem 1 (derived) helps us in finding non-zero vectors , ...., in V with
1 r
respect to T-annihilators p , p , ....p such that the vector space is a direct sum of T-invariant
1 2 r
subspaces along with a proper T-admissible subspace W .
0
 Certain concepts like complementary subspace T-admissible subspace, proper T-invariant
subspaces are explained.
 If T is a nilpotent linear operator on a vector space of dimension n, then the characteristic
n
polynomial for T is x .
 It is shown that if there is direct sum decomposition theorem 1 the cyclic ordered basis
( ,Td ,....T k 1 ) for Z ( ; T) then with the help of companion matrices, A representing T
i i i i
can be put in Jordan Form.

20.5 Keywords

Complementary Subspace: T-invariant subspace W has the property that there exists a subspace
W , such that V = W W , where W is called a complementary subspace of W. W can also be
T-invariant.
Rational Form: An n × n matrix A

A 1 0 0 ... 0
0 A 0 ... 0
2
A 0 0 ... 0
  0
0 0 ... A n

which is direct sum of companion matrices A has a rational form.
i

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