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Matrices with complex entries admit 2 fundamental operations,
The result of the composition of both these transformations is called the hermitian matrix. Matrices are defined to be unitary, iff their inverses are given by their hermitians. - Sure any purely real valued unitary matrix is therefore an orthogonal matrix. The set of unitary n x n-matrices is denoted by U(n).
The general member of U(2) is easily derived. It is given by
| e |
(
|
a b -b* a* |
)
|
where further |e| = 1 and |a|2 + |b|2 = 1.
Let A, B, C, D be non-negative reals (absolut values) and α, β, γ, δ be phases (arguments).
|
(
|
1 0 0 1 |
)
|
= |
(
|
Aeiα Beiβ Ceiγ Deiδ |
)
|
(
|
Ae-iα Ce-iγ Be-iβ De-iδ |
)
|
= |
(
|
A2 + B2 ACei(α-γ) + BDei(β-δ) ACe-i(α-γ) + BDe-i(β-δ) C2 + D2 |
)
|
Case A=0: From entry (1,1) it thus follows B=1. From (1,2) or (2,1) it then follows D=0. From (2,2) it then follows C=1. For simplicity assume further α = δ = 0 as well. Therefore:
|
(
|
Aeiα Beiβ Ceiγ Deiδ |
)
| = ei(β+γ+π)/2 |
(
|
0 ei(β-γ-π)/2 -e-i(β-γ-π)/2 0 |
)
|
Case B=0: From (1,1) it thus follows A=1. From (1,2) or (2,1) it then follows C=0. From (2,2) it then follows D=1. For simplicity assume further β = γ = 0 as well. Therefore:
|
(
|
Aeiα Beiβ Ceiγ Deiδ |
)
| = ei(α+δ)/2 |
(
|
ei(α-δ)/2 0 0 e-i(α-δ)/2 |
)
|
Cases C=0 or D=0 are identical to the formers.
Else: We thus can assume hence forward that A, B, C, D are strictely positive.
From (1,2) and (2,1) we get
-AC/BD = ei(-α+β+γ-δ) = -1.
From (1,1) and (2,2) we can assume that A=cos(φ), B=sin(φ), C=sin(ψ), D=cos(ψ) with some
0<φ<π/2 and 0<ψ<π/2.
Together this gives -tan(ψ)/tan(φ)=-1, that is φ=ψ (in this range).
So in fact we have A=D, B=C.
Further we have α-β=γ-δ+π. Therefore:
|
(
|
Aeiα Beiβ Ceiγ Deiδ |
)
| = ei(α+δ)/2 |
(
|
Aei(α-δ)/2 Bei(-α+2β-δ)/2 Bei(-α+2γ-δ)/2 Ae-i(α-δ)/2 |
)
| = ei(α+δ)/2 |
(
|
Aei(α-δ)/2 Bei(-α+2β-δ)/2 -Be-i(-α+2β-δ)/2 Ae-i(α-δ)/2 |
)
|
□
Matrices u which follow the equation un = 1 (for the first time) are said to be of order n.
The case of order 1 is trivial, u clearly is the identity matrix 1.
Order 2 unitary 2 x 2-matrices u have 2 general possibilities, eiter u = -1 or
| ± |
(
|
v c c* -v |
)
|
where further v is real, subject to v2 ≤ 1, and c is subject to |c|2 = 1 - v2.
|
(
|
1 0 0 1 |
)
| = e |
(
|
a b -b* a* |
)
| e |
(
|
a b -b* a* |
)
| = e2 |
(
|
a2-bb* (a+a*)b -(a+a*)b* a*2-bb* |
)
|
From entry (1,2) or (2,1) we have either b=0 or a+a*=0.
Case b=0: Then |a|=1 and further from (1,1) and (2,2) e2a2=e2a*2=1. Thence 0=e2(a2-a*2)=e2(a+a*)(a-a*). Cause |e|=|a|=1, hence either a is purely imaginary or purely real, i.e. a=ik for some integral k.
Subcase k even (a real): Here we have e2=1 or e=±1. But equal signs of a and e give order 1 only. Thus:
|
(
|
-1 0 0 -1 |
)
|
Subcase k odd (a imaginary): Here we have e2=-1 or e=±i. Extracting the further factor i in front of the matrix we get:
| ± |
(
|
1 0 0 -1 |
)
|
Case a+a*=0 (a imaginary): Here we can use the fact that uu=1=uu*, because we both assume order 2 and unitarity. Thus we have especially u=u*, or from entry (1,1) of u itself: ea=e*a*=e*(-a) (the latter identity by assumption of pure imaginarity of a). Therefore either a=0 or e=±i
Subcase a=0: Then |b|=1. u=u* restricted to (1,2) further gives eb=-e*b and thus leads to e=±i in this case as well. So we are left with:
| ± |
(
|
0 (ib) (ib)* 0 |
)
|
Else: Neither a nor b is zero, but a=iv for some real number v and e=±i.
| e |
(
|
a b -b* a* |
)
| = ±i |
(
|
iv b -b* -iv |
)
| = ± |
(
|
-v ib (ib)* v |
)
|
□
Order 3 (ζ3 = exp(2πi/3)): unitary 2 x 2-matrices u of this order again have 2 general possibilities, either
|
(
|
ζ3m 0 0 ζ3n |
)
|
but not both m ≡ 0 and n ≡ 0 (modulo 3) – in which case u would be the identity matrix, which clearly is of order 1 only –, or
| ζ3k |
(
|
-1/2 + iv b -b* -1/2 - iv |
)
|
where further v is real, subject to v2 ≤ 3/4, and b only is subject to bb* = 1 - aa* = 3/4 - v2.
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