8Matrix groups

IA Groups

8.5 Unitary groups

The concept of orthogonal matrices only make sense if we are talking about

real matrices. If we are talking about complex, then instead we need unitary

matrices. To do so, we replace the transposition with the Hermitian conjugate.

It is defined by

A

†

= (

A

∗

)

T

with (

A

†

)

ij

=

A

∗

ji

, where the asterisk is the complex

conjugate. We still have

(i) (AB)

†

= B

†

A

†

(ii) (A

−1

)

†

= (A

†

)

−1

(iii) A

†

A = I ⇔ AA

†

= I ⇔ A

†

= A

−1

. We say A is a unitary matrix

(iv) det A

†

= (det A)

∗

Definition (Unitary group U(n)). The unitary group is

U(n) = U

n

= {A ∈ GL

n

(C) : A

†

A = I}.

Lemma. det

: U(

n

)

→ S

1

, where

S

1

is the unit circle in the complex plane, is

a surjective group homomorphism.

Proof.

We know that 1 =

det I

=

det A

†

A

=

| det A|

2

. So

| det A|

= 1. Since

det AB = det A det B, it is a group homomorphism.

Now given

λ ∈ S

1

, we have

λ 0 · · · 0

0 1 · · · 0

.

.

.

.

.

.

.

.

.

0

0 0 0 1

∈

U(

n

). So it is surjective.

Definition

(Special unitary group

SU

(

n

))

.

The special unitary group

SU

(

n

) =

SU

n

is the kernel of det U(n) → S

1

.

Similarly, unitary matrices preserve the complex dot product: (

Ax

)

·

(

Ay

) =

x · y.