Page 293 - DMTH502_LINEAR_ALGEBRA

Unit 26: Unitary Operators and Normal Operators

be an ordered orthonormal basis for V, such that T = c , j = 1, …, n. If D = [T] , then D is the Notes
j j j 
diagonal matrix with diagonal entries c , …, c . Let P be the matrix with column vectors ,
1 n 1
–1
…, . Then D = P AP.
n
n
In case each entry of A is real, we can take V to be R , with the standard inner product, and repeat
the argument. In this case, P will be a unitary matrix with real entries, i.e., a real orthogonal
matrix.
Combining Theorem 9 with our comments at the beginning of this section, we have the following:
If V is a finite-dimensional real inner product space and T is a linear operator on V, then V has
an orthonormal basis of characteristic vectors for T if and only it T is self-adjoint. Equivalently,
t
if A is an n  n matrix with real entries, there is a real orthogonal matrix P such that P AP is
diagonal if and only if A = A . There is no such result for complex symmetric matrices. In other
t
words, for complex matrices there is a significant difference between the conditions A = A and
t
A = A*.
Having disposed of the self-adjoint case, we now return to the study of normal operators in
general. We shall prove the analogue of Theorem 9 for normal operators, in the complex case.
There is a reason for this restriction. A normal operator on a real inner product space may not
2
have any non-zero characteristic vectors. This is true, for example, of all but two rotations in R .
Theorem 10: Let V be a finite-dimensional inner product space and T a normal operator on V.
Suppose is a vector in V. Then is a characteristic vector for T with characteristic value c if and
only if is a characteristic vector for T* with characteristic value c .

Proof: Suppose U is any normal operator on V. Then U = U* . For using the condition
UU* = U*U one sees that
U 2 = (U |U ) = ( |U*U )

= ( |UU* ) = (U* |U* ) = U* 2 .
If c is any scalar, the operator U = T – cI is normal. For (T– cI)* = T* – c I, and it is easy to check that
UU* = U*U. Thus

(T – cI) = (T* – cI)
so that (T – cI) = 0 if and only if (T* – c I) = 0.
Definition: A complex n x n matrix A is called normal if AA* = A*A.
It is not so easy to understand what normality of matrices or operators really means; however,
in trying to develop some feeling for the concept, the reader might find it helpful to know that
a triangular matrix is normal if and only if it is diagonal.

Theorem 11: Let V be a finite-dimensional inner product space, T a linear operator on V, and on
orthonormal basis for V. Suppose that the matrix A of T in the basis is upper triangular. Then
T is normal if and only if A is a diagonal matrix.
Proof: Since is an orthonormal basis, A* is the matrix of T* in . If A is diagonal, then AA* =
A*A, and this implies TT* = T*T. Conversely, suppose T is normal, and let = { , . . ., }. Then,
1 n
since A is upper-triangular, T = A . By Theorem 10 this implies, T* = A . On the other
1 11 1 1 11 1
hand,

A
T* = ( *) 1 j j
1
j
= A
1 j j
j

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