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: (
A
x)
·
(
A
y) =
x · y.