Theorem 14.4 A necessary and sufficient condition for a linear transformation A on
an inner product space to be a normal operator is
kA
{ xk ¼ kAxk:
ð14:45Þ
14.3 Unitary Diagonalization of Matrices
A normal operator has a distinct property. The normal operator can be diagonalized
by a similarity transformation by a unitary matrix. The transformation is said to be a
unitary similarity transformation. Let us prove the following theorem.
Theorem 14.5 [3] A necessary and sufficient condition for a matrix A to be
diagonalized by unitary similarity transformation is that the matrix A is a normal
matrix.
Proof To prove the necessary condition, suppose that A can be diagonalized by a
unitary matrix U. That is,
U
{ AU ¼ D, i:e:, A ¼ UDU
{ and A
{
¼ UD
{ U
{ ,
ð14:46Þ
where D is a diagonal matrix. Then
AA
{
¼ UDU
{
À
Á
UD
{ U
{
À
Á ¼ UDD
{ U
{
¼ UD
{ DU
{
¼ UD
{ U
{
À
Á
UDU
{
À
Á ¼ A
{ A:
ð14:47Þ
For the third equality, we used DD
{
¼ D
{ D (i.e., D and D
{ are commutable). This
shows that A is a normal matrix.
To prove the sufficient condition, let us show that a normal matrix can be
diagonalized by unitary similarity transformation. The proof is due to mathematical
induction, as is the case with Theorem 12.1.
First we show that Theorem is true of a (2, 2) matrix. Suppose that one of
eigenvalues of A 2 is α 1 and that its corresponding eigenvector is jx 1 i. Following
procedures of the proof for Theorem 12.1 and remembering the Gram–Schmidt
orthonormalization theorem, we can construct a unitary matrix U 1 such that
U 1 ¼ jx 1 i jp 1 i
ð
Þ ,
ð14:48Þ
where jx 1 i represents a column vector and jp 1 i is another arbitrarily determined
column vector. Then we can convert A 2 to a triangle matrix such that
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14 Hermitian Operators and Unitary Operators