Page 287 - DMTH502_LINEAR_ALGEBRA

Unit 26: Unitary Operators and Normal Operators

Then A is orthogonal if and only if Notes
1 d b
t
A = A–1 = .
ad bc c a
The determinant of any orthogonal matrix is easily seen to be 1. Thus A is orthogonal if
and only if

a b
A =
b a

a b
or A =
b a
where a + b = 1. The two cases are distinguished by the value of det A.
2
2
(c) The well-known relations between the trigonometric functions show that the matrix
cos sin
A = sin cos

is orthogonal. If is a real number, then A is the matrix in the standard ordered basis for
2
R of the linear operator U , rotation through the angle . The statement that A is a real
orthogonal matrix (hence unitary) simply means that U is a unitary operator, i.e., preserves
dot products.

(d) Let
a b
A =
c d
Then A is unitary if and only if

a c 1 d b
= .
b d ad bc c a

The determinant of a unitary matrix has absolute value 1, and is thus a complex number of
the form e , real. Thus A is unitary if and only if
i
a b 1 0 a b
A = i i i
e b e a 0 e b a
2
2
where is a real number, and a, b are complex numbers such that |a| + |b| = 1.
As noted earlier, the unitary operators on an inner product space form a group. From this and
Theorem 4 it follows that the set U (n) of all n × n unitary matrices is also a group. Thus the
inverse of a unitary matrix and the product of two unitary matrices are again unitary. Of course
–1
this is easy to see directly. An n × n matrix A with complex entries is unitary if and only if A =
–1
–1 –1
–1
A*. Thus, if A is unitary, we have (A ) = A = (A*) = (A )*. If A and B are n × n unitary matrices,
–1
–1
–1
then (AB) = B A = B*A* = (AB)*.
n
The Gram-Schmidt process in C has an interesting corollary for matrices that involves the
group U (n).
Theorem 5: For every invertible complex n × n matrix B there exists a unique lower-triangular
matrix M with positive entries on the main diagonal such that MB is unitary.

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