Page 326 - DMTH502_LINEAR_ALGEBRA

Linear Algebra

Notes But this contradicts the fact that V , … V is a maximal collection of subspaces satisfying (i), (iii),
1 s
and (2). Therefore, W = V, and since
x a b
det = (x – a) + b 2
2
b x a
it follows from (i), (ii) and (iii) that
2 s
2
det (xI – T) = [(x – a) + b ] .
Corollary: Under the conditions of the theorem, T is invertible, and
–1
2
2
T* = (a + b ) T .
Proof: Since
a b a b a 2 b 2 0
b a b a = 0 a 2 b 2
it follows from (iii) and (2) that TT* = (a + b )I. Hence T is invertible and T* = (a + b )T .
2
2
2
–1
2
Theorem 11: Let T be a normal operator on a finite-dimensional inner product space V. Then any
linear operator that commutes with T also commutes with T*. Moreover, every subspace invariant
under T is also invariant under T*.
Proof: Suppose U is a linear operator on V that commutes with T. Let E, be the orthogonal
projection of V on the primary component W (1 j k) of V under T. Then E is a polynomial in
j j
T and hence commutes with U. Thus
2
EUE = UE = UE .
j j j j
Thus U(W ) is a subset of W . Let T and U denote the restrictions of T and U to W . Suppose I is the
j j j j j j
identity operator on W . Then U commutes with T , and if T = c I , it is clear that U also commutes
j j j j j j j
*
with T = c I . On the other hand, if T is not a scalar multiple of I , then T is invertible and there
j j
j
j
j
j
exist real numbers a and b such that
j j
2
*
1
T j = a 2 j b T j .
j
1
1
Since U T = T U , it follows that T U U T j . Therefore U commutes with T in both cases.
*
j j j j j j j j j
Now T* also commutes with E , and hence W is invariant under T*. Moreover for every and
j j
in W
j
(T | ) = (T | ) = ( |T* ) = ( | *T ).
j j
Since T*(W ) is contained in W , this implies T is the restriction of T* to W . Thus
*
j j j j
UT* = T*U
j j
for every in W . Since V is the sum of W , …, W , it follows that
j j 1 k
UT* = T*U
for every in V and hence that U commutes with T*.
Now suppose W is a subspace of V that is invariant under T, and let Z = W W . By the corollary
j j
to Theorem 9, W = Z j . Thus it suffices to show that each Z is invariant under *T . This is clear
j j l
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