find A matrix from one turn map matrix T such that:
[Rotx 0 0 ]
inv(A)*T*A=[ 0 Rotz 0 ]
[ 0 0 Rots]
Order it so that it is close to the order of x,y,z
also ensure that positive modes are before negative so that
one has proper symplecticity
B. Nash July 18, 2013
we order and normalize the vectors via
v_j' jmat(3) v_k = i sgn(j) delta(j,k)0001 function a=amat(transmat) 0002 %find A matrix from one turn map matrix T such that: 0003 % 0004 % [Rotx 0 0 ] 0005 %inv(A)*T*A=[ 0 Rotz 0 ] 0006 % [ 0 0 Rots] 0007 % 0008 %Order it so that it is close to the order of x,y,z 0009 %also ensure that positive modes are before negative so that 0010 %one has proper symplecticity 0011 %B. Nash July 18, 2013 0012 %we order and normalize the vectors via 0013 % v_j' jmat(3) v_k = i sgn(j) delta(j,k) 0014 0015 persistent Vxyz jm 0016 if isempty(Vxyz) 0017 Vxyz={... 0018 1/sqrt(2)*[-1i -1;1 1i;],... 0019 1/sqrt(2)*[-1i -1 0 0;1 1i 0 0;0 0 -1i -1;0 0 1 1i;],... 0020 1/sqrt(2)*[-1i -1 0 0 0 0;1 1i 0 0 0 0;0 0 -1i -1 0 0;... 0021 0 0 1 1i 0 0; 0 0 0 0 -1i -1;0 0 0 0 1 1i;]... 0022 }; 0023 jm={jmat(1),jmat(2),jmat(3)}; 0024 end 0025 0026 nv=size(transmat,1); 0027 dms=nv/2; 0028 idx=reshape(1:nv,2,dms); 0029 select=idx(1,:); 0030 0031 [V,~]=eig(transmat); 0032 0033 %compute norms of each: 0034 Vp=V'*jm{dms}; 0035 n=-0.5i*sum(Vp.'.*V); 0036 0037 %put positive modes before negative modes (swap columns if first n in pair 0038 %is negative) 0039 order=reshape(idx([1 1],:),1,nv) + (n<0); 0040 V=V(:,order); 0041 n=n(order); 0042 0043 %now normalize each vector 0044 Vn=V./repmat(sqrt(abs(n)),nv,1); 0045 0046 %find the vecs that project most onto x,y,z, and reorder 0047 %nn will have structure 0048 % n1x n1y n1z 0049 % n2x n2y n2z 0050 % n3x n3y n3z 0051 nn=0.5*abs(sqrt(-1i*Vn'*jm{dms}*Vxyz{dms})); 0052 [~,ind]=max(nn(select,select)); 0053 0054 indS=ind; 0055 %sort ind for the case of repeated indices 0056 %[~,indS]=sort(ind,2); 0057 %logic still not quite correct- should find best projector on x then y then 0058 %z. Can the one vector project best on both x and y? Is this projection 0059 %definition correct? 0060 %14 Oct, 2014- more testing needed before using this additional sort. 0061 %B. Nash 0062 0063 %reorder pairs 0064 V_ordered=Vn(:,2*indS-1); 0065 0066 %build a matrix 0067 a=reshape([real(V_ordered);imag(V_ordered)],nv,nv); 0068 0069 end 0070 0071