Page 191 - DMTH502_LINEAR_ALGEBRA

Unit 15: Simultaneous Triangulation and Simultaneous Diagonalization

Proof: It is no loss of generality to assume that  contains only a finite number of operators, Notes
because of this observation. Let {T ,...,T ) be a maximal linearly independent subset of , i.e., a
1 n
basis for the subspace spanned by . If  is a vector such that (b) holds for each T , then (b) will
i
hold for every operator which is a linear combination of T ,..., T .
1 r
By the lemma before Theorem 1 of unit 14 (this lemma for a single operator), we can find a
vector  (not in W) and a scalar c such that (T – c I) is in W. Let V be the collection of all
1 1 1 1 1 1
vectors  in V such that (T – c I) is in W. Then V is a subspace of V which is properly larger
1 1 1
than W. Furthermore, V is invariant under , for this reason. If T commutes with T , then
1 1
(T – c I)(T) = T(T – c I)
1 1 1 1
If  is in V , then (T – c I) is in W. Since W is invariant under each T in , we have T(T – c I)
1 1 1 1 1
in W, i.e., T in V , for all  in V and all T in .
1 1
Now W is a proper subspace of V . Let U be the linear operator on V obtained by restricting T
1 2 1 2
to the subspace V . The minimal polynomial for U divides the minimal polynomial for T .
1 2 2
Therefore, we may apply the lemma before Theorem 1 of unit 14 to that operator and the
invariant subspace W. We obtain a vector  in V (not in W) and a scalar c such that (T – c I) 
2 1 2 2 2 2
is in W. Note that
(a)  is not in W;
2
(b) (T – c I) is in W;
1 1 2
(c) (T – c I) is in W.
2 2 2
Let V be the set of all vectors  in V such that (T – c I) is in W. Then V is invariant under .
2 1 2 2 2
Apply the lemma before Theorem 1 of unit 14 to U , the restriction of T to V . If we continue in
3 3 2
this way, we shall reach a vector  =  (not in W) such that (T – c I) is in W, j = 1,..., r.
r j j
Theorem 1: Let V be a finite-dimensional vector space over the field F. Let  be a commuting
family of triangulable linear operators on V. There exists an ordered basis for V such that every
operator in  is represented by a triangular matrix in that basis.

Proof: Given the lemma which we just proved, this theorem has the same proof as does
Theorem 1 of unit 14, if one replaces T by .
Corollary: Let  be a commuting family of n × n matrices over an algebraically closed field F.
There exists a non-singular n × n matrix P with entries in F such that P AP is upper-triangular,
–1
for every matrix A in .
Theorem 2: Let F be a commuting family of diagonalizable linear operators on the finite-
dimensional vector space V. There exists an ordered basis for V such that every operator in  is
represented in that basis by a diagonal matrix.
Proof: We could prove this theorem by adapting the lemma before Theorem 1 to the
diagonalizable case, just as we adapted the lemma before Theorem 1 of unit 14 to the
diagonalizable case in order to prove Theorem 2 of unit 14. However, at this point it is easier to
proceed by induction on the dimension of V.

If dim V = 1, there is nothing to prove. Assume the theorem for vector spaces of dimension less
than n, and let V be an n-dimensional space. Choose any T in  which is not a scalar multiple of
the identity. Let c ,..., c be the distinct characteristic values of T, and (for each i) let W be the null
1 k i
space of T – c I. Fix an index i. Then W is invariant under every operator which commutes
i i
with T. Let  be the family of linear operators on W obtained by restricting the operators in 
i i
to the (invariant) subspace W . Each operator in  is diagonalizable, because its minimal
i i
polynomial divides the minimal polynomial for the corresponding operator in . Since dim
W < dim V, the operators in  can be simultaneously diagonalized. In other words, W has a
i i i

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