作者JohnMash (Paul)
看板Math
标题Re: [线代] 多维空间中的旋转
时间Thu Apr 21 22:04:17 2011
※ 引述《BBSealion (海狮)》之铭言：
: 我好像应该先弄懂这个问题才对XD
: 对一个N*N维的矩阵A
: 限制其为Unitary会减少几个自由度呢?
Let A=e^(iR)
A.A^H=I=e^(iR) e^(-iR^H) where M^H is the hermitian conjugate of M
hence, R=R^H
There are 2*[1+2+...+(N-1)]+N=N^2 independent real variables.
: 还有限制 det(A)=1 会减少几个自由度(这个是1吧?)
A=e^(iR)
because R=R^H is hermitian,
R=U^H.D.U
where U is unitary and D is real diagonal.
D=diag[d_1,d_2,...,d_N]
det(A)=det(e^(iU^H.D.U))=det(U^H.e^(iD).U)=det(e^(iD))
=e^(i(d_1+d_2+...+d_N))=1
d_1+d_2+...+d_N=0
Trace(D)=0
Trace(U^H.D.U)=Trace(R)=0
R is traceless.
Hence, A=e^(iR) is SU(N) if and only if R is traceless hermitian matrix
There are N^2-1 independent real variables.
--
※ 发信站: 批踢踢实业坊(ptt.cc)
◆ From: 112.104.172.100
1F：推 BBSealion :THX! 04/21 23:00
※ 编辑: JohnMash 来自: 112.104.114.174 (04/22 06:49)