Page 328 - DMTH502_LINEAR_ALGEBRA

Linear Algebra

Notes
= A U k
kj
k
= A kj k
k

Hence [T] = A = [T ] .
B B
Conversely, suppose there is an orthonormal basis  of V and an orthonormal basis  of V such
that

[T] = {T }
 
and let A = [T] . Suppose  = { , …, } and that  = { , …, }. Let U be the linear transformation
 1 n 1 n
of V into V such that U = (1 j n). Then U is a unitary transformation of V onto V , and
j j
UTU –1 = UT
j j
= U A kj k
k

= A .
kj k
k
–1
Therefore, UTU –1 = T (1 j n), and this implies UTU = T .
j j
It follows immediately from the lemma that unitarily equivalent operators on finite-dimensional
spaces have the same characteristic polynomial. For normal operators the converse is valid.
Theorem 13: Let V and V be finite-dimensional inner product spaces over the same field. Suppose
T is a normal operator on V and that T is a normal operator on V . Then T is unitarily equivalent
to T if and only if T and T have the same characteristic polynomial.
Proof: Suppose T and T have the same characteristic polynomial f. Let W (1 j k) be the
j
primary components of V under T and T the restriction of T to W . Suppose I is the identity
j j j
operator on W . Then
j
k
f = det (xI 3 T j )
j 1
Let p be the minimal polynomial for T . If p = x – c it is clear that
j j j j

det (xI – T ) = (x c ) s j
j j j
2
where s is the dimension of W . On the other hand, if p = (x – a ) + b with a , b real and b 0, then
2
j j j j j j j j
it follows from Theorem 10 that
s j
det (xI – T ) = p
j j j
s j
where in this case 2s is the dimension of W . Therefore f = p . Now we can also compute f by
j j j
j
the same method using the primary components of V under t . Since p , …, p are distinct primes,
1 k
if follows from the uniqueness of the prime factorization of f that there are exactly k primary
components W (1 j k) of V under T and that these may be indexed in such a way that p is the
j j
minimal polynomial for the restriction T of T to W . If p = x – c , then T = c I and T = c I where
j j j j j j j j j j
I is the identity operator on W . In this case it is evident that T is unitarily equivalent to T . If
j j j j
p = (x – a ) + b as above, then using the lemma and theorem 12, we again see that T is unitarily
2
2
j j j j
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