Page 187 - DMTH502_LINEAR_ALGEBRA

Unit 14: Invariant Subspaces

Proof: What (a) and (b) say is that the T-conductor of  into W is a linear polynomial. Let  be any Notes
vector in V which is not in W. Let g be the T-conductor of  into W. Then g divides p, the minimal
polynomial for T. Since  is not in W, the polynomial g is not constant. Therefore,
e
g = (x – c ) 1 ... (x – c ) k e
1 k
where at least one of the integers e is positive. Choose j so that e > 0.
i j
Then (x – c ) divides g:
j
g = (x – c )h
j
By the definition of g, the vector  = h(T) cannot be in W. But
(T – c I) = (T – c I)h(T)
j j
= g(T)

is in W.
Theorem 1: Let V be a finite-dimensional vector space over the field F and let T be a linear
operator on V. Then T is triangulable if and only if the minimal polynomial for T is a product of
linear polynomials over F.
Proof: Suppose that the minimal polynomial factors
r
p = (x – c ) 1 ... (x – c ) k r
1 k
By repeated application of the lemma above, we shall arrive at an ordered basis  = { ,..., } in
1 n
which the matrix representing T is upper triangular:

a  a a  a 1n 
11 12 13
 0 a a  a 
 22 23 2n 
T
[ ] =  0 0 a 33  a  ...(4)

3n
 
     
 0 0 0  a 
 nn
Now (4) merely says that
T =   + ... +  , 1  j  n ...(5)
j 1j 1 jj j
that is, T is in the subspace spanned by  ,...,. To find  ,..., , we start by applying the lemma
j 1 j 1 n
to the subspace W = {0}, to obtain the vector  . Then apply the lemma to W , the space spanned
1 1
by  , and we obtain  . Next apply the lemma to W , the space spanned by  and  . Continue
1 2 2 1 2
in that way. One point deserves comment. After  ,...,  have been found, it is the triangular-
1 i
type relations (5) for j = 1,..., i which ensure that the subspace spanned by  ,...,  is invariant
1 i
under T.
If T is triangulable, it is evident that the characteristic polynomial for T has the form
f = (x – c ) 1 ... (x – c ) k , c in F
d
d
1 k i
Just look at the triangular matrix (4). The diagonal entries a ,..., a are the characteristic values,
11 1n
with c repeated d times. But, if f can be so factored, so can the minimal polynomial p, because it
i i
divides f.
Corollary: Let F be an algebraically closed field, e.g., the complex number field. Every n × n
matrix over F is similar over F to a triangular matrix.
Theorem 2: Let V be a finite-dimensional vector space over the field F and let T be a linear
operator on V. Then T is diagonalizable if and only if the minimal polynomial for T has the form

LOVELY PROFESSIONAL UNIVERSITY 181