Page 327 - DMTH502_LINEAR_ALGEBRA

Unit 29: Spectral Theory and Properties of Normal Operators

if T = c I. When this is not the case, T is invertible and maps Z into and hence onto Z . Thus Notes
j j j j j
T 1 (Z ) Z and since
,
j j j
2
T j * = a 2 j b T j 1
j
it follows that T*(Z ) is contained in Z , for every j.
j j
Suppose T is a normal operator on a finite-dimensional inner product space V. Let W be a
subspace invariant under T. Then the preceding corollary shows that W is invariant under T*.
From this it follows that W is invariant under T** = T (and hence under T* as well). Using this
fact one can easily prove the following strengthened version of the cyclic decomposition theorem.
Theorem 12: Let T be a normal linear operator on a finite-dimensional inner product space V
(dim V 1). Then there exist r non-zero vectors , …, in V with respective T-annihilators e ,
1 r 1
…, e such that
r
(i) V = Z ( ; T) … Z( ; T);
1 r
(ii) if 1 k r – 1, then e divides e ;
k+1 k
(iii) Z( ; T) is orthogonal to Z( ; T) when j k. Furthermore, the integer r and the annihilators
j k
e , …, e are uniquely determined by conditions (i) and (ii) and the fact that no is 0.
1 r k
Corollary: If A is a normal matrix with real (complex) entries, then there is a real orthogonal
(unitary) matrix P such that P AP is in rational canonical form.
–1
It follows that two normal matrices A and B are unitarily equivalent if and only if they have the
same rational form; A and B are orthogonally equivalent if they have real entries and the same
rational form.
On the other hand, there is a simpler criterion for the unitary equivalence of normal matrices
and normal operators.
Definition: Let V and V be inner product spaces over the same field. A linear transformation
U : V V

is called a unitary transformation if it maps V onto V and preserves inner products. If T is a
linear operator on V and T a linear operator on V , then T is unitarily equivalent to T if there
exists a unitary transformation U of V onto V such that
UTU –1 = T .
Lemma: Let V and V be finite-dimensional inner product spaces over the same field. Suppose T
is a linear operator on V and that T is a linear operator on V . Then T is unitarily equivalent to
T if and only if there is an orthonormal basis  of V and an orthonormal basis  of V such that
[T] = [T ] .
 
–1
Proof: Suppose there is a unitary transformation U of V onto V such that UTU = T . Let  =
{ , …, } be any (ordered) orthonormal basis for V. Let = U (1 j n). Then  = { , …, }
1 n j j 1 n
is an orthonormal basis for V and setting
n
T = A
j kj k
k 1
we see that
T = UT
j j

LOVELY PROFESSIONAL UNIVERSITY 321