Show your support by donating any amount. (Note: We are still technically a for-profit company, so your
contribution is not tax-deductible.)
PayPal Acct: 
Feedback: 
Donate to VoyForums (PayPal):
| [ Login ] [ Contact Forum Admin ] [ Main index ] [ Post a new message ] [ Search | Check update time | Archives: 1, [2], 3, 4 ] |
s=cell(1,3);
s{1}=[complex(0,0),complex(1,0);complex(1,0),complex(0,0)];
s{1}=[complex(0,0),complex(1,0);complex(1,0),complex(0,0)]
s =
[2x2 double] [] []
s{1}=[complex(0,0),complex(1,0);complex(1,0),complex(0,0)]; s{1}
ans =
0 1
1 0
s{2}=[complex(0,0),complex(0,-1);complex(0,1),complex(0,0)]; s{2}
ans =
0 0 - 1.0000i
0 + 1.0000i 0
s{3}=[complex(1,0),complex(0,0);complex(0,0),complex(-1,0)]; s{3}
ans =
1 0
0 -1
syms n n1 n2 n3 real;
ns=sym(zeros(2)); n=[n1,n2,n3];
for k=1:3; ns=ns+n(k)*s{k}; end; ns
ns =
[ n3, n1-i*n2]
[ n1+i*n2, -n3]
[v d]=eig(ns);
v=simple(v);
pretty(v), d
[ 2 2 2 1/2 2 2 2 1/2 ]
[(n3~ + n1~ + n2~ ) + n3~ -(n3~ + n1~ + n2~ ) + n3~]
[----------------------------- ------------------------------]
[ n1~ + n2~ I n1~ + n2~ I ]
[ ]
[ 1 1 ]
d =
[ (n3^2+n1^2+n2^2)^(1/2), 0]
[ 0, -(n3^2+n1^2+n2^2)^(1/2)]
v=subs(v,'(n3^2+n1^2+n2^2)^(1/2)',1); d=subs(d,'(n3^2+n1^2+n2^2)^(1/2)',1);
pretty(v), d
[ 1 + n3~ -1 + n3~ ]
[----------- -----------]
[n1~ + n2~ I n1~ + n2~ I]
[ ]
[ 1 1 ]
d =
1 0
0 -1
% have now got simplified matrix of eigen vectors and eigenvalues,
% and have substituted in the condition that n is a unit vector.
% Now create a cell of two column vectors to take in the columns of v.
ev=cell(1,2); ev{1}=v(:,1); ev{2}=v(:,2);
pretty(ev{1}), pretty(ev{2})
[ 1 + n3~ ]
[-----------]
[n1~ + n2~ I]
[ ]
[ 1 ]
[ -1 + n3~ ]
[-----------]
[n1~ + n2~ I]
[ ]
[ 1 ]
% Let us check that these two are indeed orthogonal, as they should be.
ev{1}*ev{2}
??? Error using ==> sym/mtimes
Inner matrix dimensions must agree.
size(ev{1}), size(ev{2})
ans =
2 1
ans =
2 1
ev{1}, ev{2}
ans =
[ (1+n3)/(n1+n2*i)]
[ 1]
ans =
[ (-1+n3)/(n1+n2*i)]
[ 1]
ev{1}(1,1)
ans =
(1+n3)/(n1+n2*i)
ev{1}*ev{2}
??? Error using ==> sym/mtimes
Inner matrix dimensions must agree.
%but they do don't they?
%Looks like I'll have to do this the long way this time!
(ev{1}(1,1)*ev{2}(1,1))+(ev{1}(2,1)*ev{2}(2,1))
ans =
(1+n3)/(n1+n2*i)^2*(-1+n3)+1
pretty(ans)
(1 + n3~) (-1 + n3~)
-------------------- + 1
2
(n1~ + n2~ I)
simplify(ans)
ans =
(-1+n3^2+n1^2+2*i*n1*n2-n2^2)/(n1+n2*i)^2
pretty(ans)
2 2 2
-1 + n3~ + n1~ + 2 I n1~ n2~ - n2~
-------------------------------------
2
(n1~ + n2~ I)
% Hmmm, eigenvectors don't appear to be orthogonal
% odd...
doc eig
Overloaded methods
doc symbolic/eig
%so if the eigenvectors are not orthogonal, this suggests that the
%origonal operator, ns, is not hermition. Let us check this
ns, ns'
ns =
[ n3, n1-i*n2]
[ n1+i*n2, -n3]
ans =
[ n3, n1-i*n2]
[ n1+n2*i, -n3]
ns==ns'
ans =
1 1
0 1
n1+i*n2==n1+n2*i
ans =
1
% So, ns IS hermitian, therefore the eigenvectors have to be orthogonal
% Furthermore something funny is going on!
% I shall recalculate the eigenvectors and re-test their orthogonality
% without simplifying or substituting them
[v d]=eig(ns);
e1=v(:,1); e2=v(:,2);
e1*e2
??? Error using ==> sym/mtimes
Inner matrix dimensions must agree.
% That damn error again!!!!
size(e1), size(e2)
ans =
2 1
ans =
2 1
size(e1)==size(e2)
ans =
1 1
e1
e1 =
[ (n1*(n3^2+n1^2+n2^2)^(1/2)+n1*n3-i*n2*(n3^2+n1^2+n2^2)^(1/2)-i*n2*n3)/(n1^2+n2^2)]
[ 1]
e2
e2 =
[ (-n1*(n3^2+n1^2+n2^2)^(1/2)+n1*n3+i*n2*(n3^2+n1^2+n2^2)^(1/2)-i*n2*n3)/(n1^2+n2^2)]
[ 1]
e1(1)*e2(1)+e1(2)*e2(2)
ans =
(n1*(n3^2+n1^2+n2^2)^(1/2)+n1*n3-i*n2*(n3^2+n1^2+n2^2)^(1/2)-i*n2*n3)/(n1^2+n2^2)^2*(-n1*(n3^2+n1^2+n2^2)^(1/2)+n1*n3+i*n2*(n3^2+n1^2+n2^2)^(1/2)-i*n2*n3)+1
pretty(ans)
1/2 1/2
(n1~ %1 + n1~ n3~ - n2~ %1 I - n2~ n3~ I)
1/2 1/2 / 2 2 2
(-n1~ %1 + n1~ n3~ + n2~ %1 I - n2~ n3~ I) / (n1~ + n2~ )
/
+ 1
2 2 2
%1 := n3~ + n1~ + n2~
simple(ans)
simplify:
2*n2*(i*n1+n2)/(n1^2+n2^2)
radsimp:
2*n2*(i*n1+n2)/(n1^2+n2^2)
combine(trig):
(2*i*n1*n2+2*n2^2)/(n1^2+n2^2)
factor:
2*i*n2/(n1+n2*i)
expand:
-1/(n1^2+n2^2)^2*n1^4+2*i/(n1^2+n2^2)^2*n1^3*n2+2*i/(n1^2+n2^2)^2*n1*n2^3+1/(n1^2+n2^2)^2*n2^4+1
combine:
(n1*(n3^2+n1^2+n2^2)^(1/2)+n1*n3-i*n2*(n3^2+n1^2+n2^2)^(1/2)-i*n2*n3)/(n1^2+n2^2)^2*(-n1*(n3^2+n1^2+n2^2)^(1/2)+n1*n3+n2*(n3^2+n1^2+n2^2)^(1/2)*i-i*n2*n3)+1
convert(exp):
(n1*(n3^2+n1^2+n2^2)^(1/2)+n1*n3-i*n2*(n3^2+n1^2+n2^2)^(1/2)-i*n2*n3)/(n1^2+n2^2)^2*(-n1*(n3^2+n1^2+n2^2)^(1/2)+n1*n3+n2*(n3^2+n1^2+n2^2)^(1/2)*i-i*n2*n3)+1
convert(sincos):
(n1*(n3^2+n1^2+n2^2)^(1/2)+n1*n3-i*n2*(n3^2+n1^2+n2^2)^(1/2)-i*n2*n3)/(n1^2+n2^2)^2*(-n1*(n3^2+n1^2+n2^2)^(1/2)+n1*n3+n2*(n3^2+n1^2+n2^2)^(1/2)*i-i*n2*n3)+1
convert(tan):
(n1*(n3^2+n1^2+n2^2)^(1/2)+n1*n3-i*n2*(n3^2+n1^2+n2^2)^(1/2)-i*n2*n3)/(n1^2+n2^2)^2*(-n1*(n3^2+n1^2+n2^2)^(1/2)+n1*n3+n2*(n3^2+n1^2+n2^2)^(1/2)*i-i*n2*n3)+1
collect(n1):
(n1*(n3^2+n1^2+n2^2)^(1/2)+n1*n3-i*n2*(n3^2+n1^2+n2^2)^(1/2)-i*n2*n3)/(n1^2+n2^2)^2*(-n1*(n3^2+n1^2+n2^2)^(1/2)+n1*n3+n2*(n3^2+n1^2+n2^2)^(1/2)*i-i*n2*n3)+1
mwcos2sin:
(n1*(n3^2+n1^2+n2^2)^(1/2)+n1*n3-i*n2*(n3^2+n1^2+n2^2)^(1/2)-i*n2*n3)/(n1^2+n2^2)^2*(-n1*(n3^2+n1^2+n2^2)^(1/2)+n1*n3+n2*(n3^2+n1^2+n2^2)^(1/2)*i-i*n2*n3)+1
ans =
2*i*n2/(n1+n2*i)
pretty(ans)
2 I n2~
-----------
n1~ + n2~ I
dim
??? Undefined function or variable 'dim'.
dimension
??? Undefined function or variable 'dimension'.
% I realise I should be multiplying by the conjugate of one of the vectors!
e1*e2'
ans =
[ (n1*(n3^2+n1^2+n2^2)^(1/2)+n1*n3-i*n2*(n3^2+n1^2+n2^2)^(1/2)-i*n2*n3)/(n1^2+n2^2)^2*(-n1*(n3^2+n1^2+n2^2)^(1/2)+n1*n3-i*n2*(n3^2+n1^2+n2^2)^(1/2)+i*n2*n3), (n1*(n3^2+n1^2+n2^2)^(1/2)+n1*n3-i*n2*(n3^2+n1^2+n2^2)^(1/2)-i*n2*n3)/(n1^2+n2^2)]
[ (-n1*(n3^2+n1^2+n2^2)^(1/2)+n1*n3-i*n2*(n3^2+n1^2+n2^2)^(1/2)+i*n2*n3)/(n1^2+n2^2), 1]
% wrong again!
e1'*e2
ans =
(n1*(n3^2+n1^2+n2^2)^(1/2)+n1*n3+n2*(n3^2+n1^2+n2^2)^(1/2)*i+i*n2*n3)/(n1^2+n2^2)^2*(-n1*(n3^2+n1^2+n2^2)^(1/2)+n1*n3+n2*(n3^2+n1^2+n2^2)^(1/2)*i-i*n2*n3)+1
simple(ans)
simplify:
0
radsimp:
0
combine(trig):
0
factor:
0
expand:
-1/(n1^2+n2^2)^2*n1^4-2/(n1^2+n2^2)^2*n1^2*n2^2-1/(n1^2+n2^2)^2*n2^4+1
combine:
(n1*(n3^2+n1^2+n2^2)^(1/2)+n1*n3+n2*(n3^2+n1^2+n2^2)^(1/2)*i+i*n2*n3)/(n1^2+n2^2)^2*(-n1*(n3^2+n1^2+n2^2)^(1/2)+n1*n3+n2*(n3^2+n1^2+n2^2)^(1/2)*i-i*n2*n3)+1
convert(exp):
(n1*(n3^2+n1^2+n2^2)^(1/2)+n1*n3+n2*(n3^2+n1^2+n2^2)^(1/2)*i+i*n2*n3)/(n1^2+n2^2)^2*(-n1*(n3^2+n1^2+n2^2)^(1/2)+n1*n3+n2*(n3^2+n1^2+n2^2)^(1/2)*i-i*n2*n3)+1
convert(sincos):
(n1*(n3^2+n1^2+n2^2)^(1/2)+n1*n3+n2*(n3^2+n1^2+n2^2)^(1/2)*i+i*n2*n3)/(n1^2+n2^2)^2*(-n1*(n3^2+n1^2+n2^2)^(1/2)+n1*n3+n2*(n3^2+n1^2+n2^2)^(1/2)*i-i*n2*n3)+1
convert(tan):
(n1*(n3^2+n1^2+n2^2)^(1/2)+n1*n3+n2*(n3^2+n1^2+n2^2)^(1/2)*i+i*n2*n3)/(n1^2+n2^2)^2*(-n1*(n3^2+n1^2+n2^2)^(1/2)+n1*n3+n2*(n3^2+n1^2+n2^2)^(1/2)*i-i*n2*n3)+1
collect(n1):
(n1*(n3^2+n1^2+n2^2)^(1/2)+n1*n3+n2*(n3^2+n1^2+n2^2)^(1/2)*i+i*n2*n3)/(n1^2+n2^2)^2*(-n1*(n3^2+n1^2+n2^2)^(1/2)+n1*n3+n2*(n3^2+n1^2+n2^2)^(1/2)*i-i*n2*n3)+1
mwcos2sin:
(n1*(n3^2+n1^2+n2^2)^(1/2)+n1*n3+n2*(n3^2+n1^2+n2^2)^(1/2)*i+i*n2*n3)/(n1^2+n2^2)^2*(-n1*(n3^2+n1^2+n2^2)^(1/2)+n1*n3+n2*(n3^2+n1^2+n2^2)^(1/2)*i-i*n2*n3)+1
ans =
0
% YES! I have been forgetting my matrix multiplication!!!
ev{1}
ans =
[ (1+n3)/(n1+n2*i)]
[ 1]
ev{1}'
ans =
[ (1+n3)/(n1-i*n2), 1]
ev{1}'*ev{2}
ans =
(1+n3)/(n1-i*n2)*(-1+n3)/(n1+n2*i)+1
simple(ans)
simplify:
(-1+n3^2+n1^2+n2^2)/(n1^2+n2^2)
radsimp:
(-1+n3^2+n1^2+n2^2)/(n1-i*n2)/(n1+n2*i)
combine(trig):
(-1+n3^2+n1^2+n2^2)/(n1^2+n2^2)
factor:
(-1+n3^2+n1^2+n2^2)/(n1-n2*i)/(n1+n2*i)
expand:
-1/(n1-i*n2)/(n1+n2*i)+1/(n1-i*n2)/(n1+n2*i)*n3^2+1
combine:
(1+n3)/(n1-i*n2)*(-1+n3)/(n1+n2*i)+1
convert(exp):
(1+n3)/(n1-i*n2)*(-1+n3)/(n1+n2*i)+1
convert(sincos):
(1+n3)/(n1-i*n2)*(-1+n3)/(n1+n2*i)+1
convert(tan):
(1+n3)/(n1-i*n2)*(-1+n3)/(n1+n2*i)+1
collect(n1):
(1+n3)/(n1-i*n2)*(-1+n3)/(n1+n2*i)+1
mwcos2sin:
(1+n3)/(n1-i*n2)*(-1+n3)/(n1+n2*i)+1
ans =
(-1+n3^2+n1^2+n2^2)/(n1^2+n2^2)
subs(ans,'n3^2+n1^2+n2^2',1)
ans =
(-n3^2-n1^2-n2^2+n3^2*(n3^2+n1^2+n2^2)+n1^2*(n3^2+n1^2+n2^2)+n2^2*(n3^2+n1^2+n2^2))^(n3^2+n1^2+n2^2)/(n1^2*(n3^2+n1^2+n2^2)+n2^2*(n3^2+n1^2+n2^2))
simple(ans)
simplify:
(-n3^2-n1^2-n2^2+n3^4+2*n3^2*n1^2+2*n3^2*n2^2+n1^4+2*n1^2*n2^2+n2^4)^(n3^2+n1^2+n2^2)/(n3^2+n1^2+n2^2)/(n1^2+n2^2)
radsimp:
(-n3^2-n1^2-n2^2+n3^4+2*n3^2*n1^2+2*n3^2*n2^2+n1^4+2*n1^2*n2^2+n2^4)^(n3^2+n1^2+n2^2)/(n3^2+n1^2+n2^2)/(n1^2+n2^2)
combine(trig):
(-n3^2-n1^2-n2^2+n3^4+2*n3^2*n1^2+2*n3^2*n2^2+n1^4+2*n1^2*n2^2+n2^4)^(n3^2+n1^2+n2^2)/(n3^2*n1^2+n1^4+2*n1^2*n2^2+n3^2*n2^2+n2^4)
factor:
((n3^2+n1^2+n2^2)*(-1+n3^2+n1^2+n2^2))^(n3^2+n1^2+n2^2)/(n3^2+n1^2+n2^2)/(n1^2+n2^2)
expand:
(-n3^2-n1^2-n2^2+n3^4+2*n3^2*n1^2+2*n3^2*n2^2+n1^4+2*n1^2*n2^2+n2^4)^(n3^2)*(-n3^2-n1^2-n2^2+n3^4+2*n3^2*n1^2+2*n3^2*n2^2+n1^4+2*n1^2*n2^2+n2^4)^(n1^2)*(-n3^2-n1^2-n2^2+n3^4+2*n3^2*n1^2+2*n3^2*n2^2+n1^4+2*n1^2*n2^2+n2^4)^(n2^2)/(n3^2*n1^2+n1^4+2*n1^2*n2^2+n3^2*n2^2+n2^4)
combine:
(-n3^2-n1^2-n2^2+n3^2*(n3^2+n1^2+n2^2)+n1^2*(n3^2+n1^2+n2^2)+n2^2*(n3^2+n1^2+n2^2))^(n3^2+n1^2+n2^2)/(n1^2*(n3^2+n1^2+n2^2)+n2^2*(n3^2+n1^2+n2^2))
convert(exp):
(-n3^2-n1^2-n2^2+n3^2*(n3^2+n1^2+n2^2)+n1^2*(n3^2+n1^2+n2^2)+n2^2*(n3^2+n1^2+n2^2))^(n3^2+n1^2+n2^2)/(n1^2*(n3^2+n1^2+n2^2)+n2^2*(n3^2+n1^2+n2^2))
convert(sincos):
(-n3^2-n1^2-n2^2+n3^2*(n3^2+n1^2+n2^2)+n1^2*(n3^2+n1^2+n2^2)+n2^2*(n3^2+n1^2+n2^2))^(n3^2+n1^2+n2^2)/(n1^2*(n3^2+n1^2+n2^2)+n2^2*(n3^2+n1^2+n2^2))
convert(tan):
(-n3^2-n1^2-n2^2+n3^2*(n3^2+n1^2+n2^2)+n1^2*(n3^2+n1^2+n2^2)+n2^2*(n3^2+n1^2+n2^2))^(n3^2+n1^2+n2^2)/(n1^2*(n3^2+n1^2+n2^2)+n2^2*(n3^2+n1^2+n2^2))
collect(n1):
(-n3^2-n1^2-n2^2+n3^2*(n3^2+n1^2+n2^2)+n1^2*(n3^2+n1^2+n2^2)+n2^2*(n3^2+n1^2+n2^2))^(n3^2+n1^2+n2^2)/(n1^4+(n3^2+2*n2^2)*n1^2+n2^2*(n3^2+n2^2))
mwcos2sin:
(-n3^2-n1^2-n2^2+n3^2*(n3^2+n1^2+n2^2)+n1^2*(n3^2+n1^2+n2^2)+n2^2*(n3^2+n1^2+n2^2))^(n3^2+n1^2+n2^2)/(n1^2*(n3^2+n1^2+n2^2)+n2^2*(n3^2+n1^2+n2^2))
ans =
((n3^2+n1^2+n2^2)*(-1+n3^2+n1^2+n2^2))^(n3^2+n1^2+n2^2)/(n3^2+n1^2+n2^2)/(n1^2+n2^2)
% Was almost there to proving ev{1} and ev{2} are orthogonal!
ev{1}'*ev{2}
ans =
(1+n3)/(n1-i*n2)*(-1+n3)/(n1+n2*i)+1
ip=simple(ans)
ip =
(-1+n3^2+n1^2+n2^2)/(n1^2+n2^2)
% n3^2+n1^2+n2^2=1 though!
ip=subs(ans,'n3^2+n1^2+n2^2',1)
ip =
((n3^2+n1^2+n2^2)^2+n3*(n3^2+n1^2+n2^2))^(n3^2+n1^2+n2^2)/(n1*(n3^2+n1^2+n2^2)+(-i)^(n3^2+n1^2+n2^2)*n2^(n3^2+n1^2+n2^2)*(n3^2+n1^2+n2^2))*(-n3^2-n1^2-n2^2+n3*(n3^2+n1^2+n2^2))^(n3^2+n1^2+n2^2)/(n1*(n3^2+n1^2+n2^2)+n2^(n3^2+n1^2+n2^2)*(i*(n3^2+n1^2+n2^2))^(n3^2+n1^2+n2^2)*(n3^2+n1^2+n2^2))*(n3^2+n1^2+n2^2)+(n3^2+n1^2+n2^2)^2
pretty(ip)
2 %1 2 2 2 %1
(%1 + n3~ %1) (-n3~ - n1~ - n2~ + n3~ %1) %1 2
------------------------------------------------------- + %1
%1 %1 %1 %1
(n1~ %1 + (-I) n2~ %1) (n1~ %1 + n2~ (%1 I) %1)
2 2 2
%1 := n3~ + n1~ + n2~
%????????????????????????
ip=subs(ans,'n3^2+n1^2+n2^2',1)
ip =
((n3^2+n1^2+n2^2)^2+n3*(n3^2+n1^2+n2^2))^(n3^2+n1^2+n2^2)/(n1*(n3^2+n1^2+n2^2)+(-i)^(n3^2+n1^2+n2^2)*n2^(n3^2+n1^2+n2^2)*(n3^2+n1^2+n2^2))*(-n3^2-n1^2-n2^2+n3*(n3^2+n1^2+n2^2))^(n3^2+n1^2+n2^2)/(n1*(n3^2+n1^2+n2^2)+n2^(n3^2+n1^2+n2^2)*((n3^2+n1^2+n2^2)*i)^(n3^2+n1^2+n2^2)*(n3^2+n1^2+n2^2))*(n3^2+n1^2+n2^2)+(n3^2+n1^2+n2^2)^2
ip=subs(ans,'n3^2+n1^2+n2^2',1)
ip =
((n3^2+n1^2+n2^2)^2+n3*(n3^2+n1^2+n2^2))^(n3^2+n1^2+n2^2)/(n1*(n3^2+n1^2+n2^2)+(-i)^(n3^2+n1^2+n2^2)*n2^(n3^2+n1^2+n2^2)*(n3^2+n1^2+n2^2))*(-n3^2-n1^2-n2^2+n3*(n3^2+n1^2+n2^2))^(n3^2+n1^2+n2^2)/(n1*(n3^2+n1^2+n2^2)+n2^(n3^2+n1^2+n2^2)*((n3^2+n1^2+n2^2)*i)^(n3^2+n1^2+n2^2)*(n3^2+n1^2+n2^2))*(n3^2+n1^2+n2^2)+(n3^2+n1^2+n2^2)^2
pertty(ip)
??? Undefined function or variable 'pertty'.
pretty(ip)
2 %1 2 2 2 %1
(%1 + n3~ %1) (-n3~ - n1~ - n2~ + n3~ %1) %1 2
------------------------------------------------------- + %1
%1 %1 %1 %1
(n1~ %1 + (-I) n2~ %1) (n1~ %1 + n2~ (%1 I) %1)
2 2 2
%1 := n3~ + n1~ + n2~
% Well, without knowledge of how to make 'smarter' substitutions,
% I think it is safe to say that I have shown that the eigenvectors
% are also orthogonal
ip=simple(ev{1}'*ev{2})
ip =
(-1+n3^2+n1^2+n2^2)/(n1^2+n2^2)
pretty(ip)
2 2 2
-1 + n3~ + n1~ + n2~
-----------------------
2 2
n1~ + n2~
% sum(ni^2)=1, therefore sum(ni^2)-1=0
% So I do have orthogonal eigenvectors, but I still have not
% found a smart way of telling matlab that their sum of square
% is equal to one!
% Perhaps if I don't get anyway on this line of attack I should
% try rewriting n in terms of spherical polar coordinates with
% a unit radius, thus reducing parameters down from 3 to 2.
% Next phase, let's normalise the (orthogonal) eigenvectors!
% and I'm not going to bother with using cell arrays either,
% nice idea tying the vectors together into a unified structure.
% However, I think for what Im' donig it's a bit redundant.
ev1=ev{1}; ev2=ev{2};
norm(ev1), norm(2)
??? Error using ==> norm
Function 'norm' is not defined for values of class 'sym'.
norm(ev1), norm(ev2)
??? Error using ==> norm
Function 'norm' is not defined for values of class 'sym'.
% I have to say, using symbolic maths presents lots of obsticals to you
% in matlab, there is probably something in TMW file exchange, but I think
% it will be quicker to fashion my own algorithm than go looking for something
% just now. I'll log in and look later.
ev1norm=sqrt(ev1(1)^2+ev1(2)^2)
ev1norm =
((1+n3)^2/(n1+n2*i)^2+1)^(1/2)
pretty(ev1norm)
/ 2 \1/2
| (1 + n3~) |
|-------------- + 1|
| 2 |
\(n1~ + n2~ I) /
ev2norm=sqrt(ev2(1)^2+ev2(2)^2); pretty(ev2norm)
/ 2 \1/2
| (-1 + n3~) |
|-------------- + 1|
| 2 |
\(n1~ + n2~ I) /
ev1normsq=ev1(1)^2+ev1(2)^2; ev2normsq=ev2(1)^2+ev2(2)^2;
% I decided it would be more convenient to eliminate the
% square root as I will be using the outer product notation
% when I compute my eigenspace projector.
% Next phase, computer the projector onto the eigen space
% To do this we need an orthonormal basis of the eigenspace.
% Hence the need for orthonormal eigenvectors. We will then
% Compute the projector in the outer-product notation...
% P=|e1>
% dim(eigenspace)=2).
eigenprojector=((ev1*ev1')/ev1normsq)+((ev2*ev2')/ev2normsq)
eigenprojector =
[ (1+n3)^2/(n1+n2*i)/(n1-i*n2)/((1+n3)^2/(n1+n2*i)^2+1)+(-1+n3)^2/(n1+n2*i)/(n1-i*n2)/((-1+n3)^2/(n1+n2*i)^2+1), (1+n3)/(n1+n2*i)/((1+n3)^2/(n1+n2*i)^2+1)+(-1+n3)/(n1+n2*i)/((-1+n3)^2/(n1+n2*i)^2+1)]
[ (1+n3)/(n1-i*n2)/((1+n3)^2/(n1+n2*i)^2+1)+(-1+n3)/(n1-i*n2)/((-1+n3)^2/(n1+n2*i)^2+1), 1/((1+n3)^2/(n1+n2*i)^2+1)+1/((-1+n3)^2/(n1+n2*i)^2+1)]
pretty(eigenprojector)
[ 2
[ (1 + n3~)
[------------------------------------------------
[ / 2 \
[ | (1 + n3~) |
[(n1~ + n2~ I) (n1~ - n2~ I) |-------------- + 1|
[ | 2 |
[ \(n1~ + n2~ I) /
2
(-1 + n3~)
+ ------------------------------------------------ ,
/ 2 \
| (-1 + n3~) |
(n1~ + n2~ I) (n1~ - n2~ I) |-------------- + 1|
| 2 |
\(n1~ + n2~ I) /
1 + n3~
----------------------------------
/ 2 \
| (1 + n3~) |
(n1~ + n2~ I) |-------------- + 1|
| 2 |
\(n1~ + n2~ I) /
]
-1 + n3~ ]
+ ----------------------------------]
/ 2 \]
| (-1 + n3~) |]
(n1~ + n2~ I) |-------------- + 1|]
| 2 |]
\(n1~ + n2~ I) /]
[ 1 + n3~
[----------------------------------
[ / 2 \
[ | (1 + n3~) |
[(n1~ - n2~ I) |-------------- + 1|
[ | 2 |
[ \(n1~ + n2~ I) /
-1 + n3~
+ ---------------------------------- ,
/ 2 \
| (-1 + n3~) |
(n1~ - n2~ I) |-------------- + 1|
| 2 |
\(n1~ + n2~ I) /
1 1 ]
------------------ + ------------------]
2 2 ]
(1 + n3~) (-1 + n3~) ]
-------------- + 1 -------------- + 1]
2 2 ]
(n1~ + n2~ I) (n1~ + n2~ I) ]
% Yikes! Lets see if we can cut this down a bit...
test=simple(eigenprojector)
test =
[ (1+n3)^2/(n1+n2*i)/(n1-i*n2)/((1+n3)^2/(n1+n2*i)^2+1)+(-1+n3)^2/(n1+n2*i)/(n1-i*n2)/((-1+n3)^2/(n1+n2*i)^2+1), (1+n3)/(n1+n2*i)/((1+n3)^2/(n1+n2*i)^2+1)+(-1+n3)/(n1+n2*i)/((-1+n3)^2/(n1+n2*i)^2+1)]
[ (1+n3)/(n1-i*n2)/((1+n3)^2/(n1+n2*i)^2+1)+(-1+n3)/(n1-i*n2)/((-1+n3)^2/(n1+n2*i)^2+1), 1/((1+n3)^2/(n1+n2*i)^2+1)+1/((-1+n3)^2/(n1+n2*i)^2+1)]
pretty(test)
[ 2
[ (1 + n3~)
[------------------------------------------------
[ / 2 \
[ | (1 + n3~) |
[(n1~ + n2~ I) (n1~ - n2~ I) |-------------- + 1|
[ | 2 |
[ \(n1~ + n2~ I) /
2
(-1 + n3~)
+ ------------------------------------------------ ,
/ 2 \
| (-1 + n3~) |
(n1~ + n2~ I) (n1~ - n2~ I) |-------------- + 1|
| 2 |
\(n1~ + n2~ I) /
1 + n3~
----------------------------------
/ 2 \
| (1 + n3~) |
(n1~ + n2~ I) |-------------- + 1|
| 2 |
\(n1~ + n2~ I) /
]
-1 + n3~ ]
+ ----------------------------------]
/ 2 \]
| (-1 + n3~) |]
(n1~ + n2~ I) |-------------- + 1|]
| 2 |]
\(n1~ + n2~ I) /]
[ 1 + n3~
[----------------------------------
[ / 2 \
[ | (1 + n3~) |
[(n1~ - n2~ I) |-------------- + 1|
[ | 2 |
[ \(n1~ + n2~ I) /
-1 + n3~
+ ---------------------------------- ,
/ 2 \
| (-1 + n3~) |
(n1~ - n2~ I) |-------------- + 1|
| 2 |
\(n1~ + n2~ I) /
1 1 ]
------------------ + ------------------]
2 2 ]
(1 + n3~) (-1 + n3~) ]
-------------- + 1 -------------- + 1]
2 2 ]
(n1~ + n2~ I) (n1~ + n2~ I) ]
% That didn't help!
test=simplify(eigenprojector)
test =
[ 2*(1-2*n3^2+n1^2+2*i*n1*n2-n2^2+n3^4+n3^2*n1^2+2*i*n3^2*n1*n2-n3^2*n2^2)*(n1+n2*i)/(n1-n2*i)/(1+2*n3+n3^2+n1^2+2*i*n1*n2-n2^2)/(1-2*n3+n3^2+n1^2+2*i*n1*n2-n2^2), 2*(-1+n3^2+n1^2+2*i*n1*n2-n2^2)*n3*(n1+n2*i)/(1-2*n3+n3^2+n1^2+2*i*n1*n2-n2^2)/(1+2*n3+n3^2+n1^2+2*i*n1*n2-n2^2)]
[ 2*(n1+n2*i)^2*n3*(-1+n3^2+n1^2+2*i*n1*n2-n2^2)/(n1-n2*i)/(1+2*n3+n3^2+n1^2+2*i*n1*n2-n2^2)/(1-2*n3+n3^2+n1^2+2*i*n1*n2-n2^2), 2*(n1+n2*i)^2*(1+n3^2+n1^2+2*i*n1*n2-n2^2)/(1-2*n3+n3^2+n1^2+2*i*n1*n2-n2^2)/(1+2*n3+n3^2+n1^2+2*i*n1*n2-n2^2)]
test(factor(eigenprojector))
??? Unable to find subsindex function for class sym.
test=factor(eigenprojector)
test =
[ 2*(1-2*n3^2+n1^2+2*i*n1*n2-n2^2+n3^4+n3^2*n1^2+2*i*n3^2*n1*n2-n3^2*n2^2)*(n1+n2*i)/(n1-i+n2*i-i*n3)/(n1-i+i*n3+n2*i)/(n1+i-i*n3+n2*i)/(n1+i+n2*i+i*n3)/(n1-n2*i), 2*(-1+n3^2+n1^2+2*i*n1*n2-n2^2)*n3*(n1+n2*i)/(n1-i+n2*i-i*n3)/(n1-i+i*n3+n2*i)/(n1+i-i*n3+n2*i)/(n1+i+n2*i+i*n3)]
[ 2*(n1+n2*i)^2*n3*(-1+n3^2+n1^2+2*i*n1*n2-n2^2)/(n1-i+n2*i-i*n3)/(n1-i+i*n3+n2*i)/(n1+i-i*n3+n2*i)/(n1+i+n2*i+i*n3)/(n1-n2*i), 2*(n1+n2*i)^2*(1+n3^2+n1^2+2*i*n1*n2-n2^2)/(n1-i+n2*i-i*n3)/(n1-i+i*n3+n2*i)/(n1+i-i*n3+n2*i)/(n1+i+n2*i+i*n3)]
pretty(test)
[
[ 2 2 2 4 2 2
[2 (1 - 2 n3~ + n1~ + 2 I n1~ n2~ - n2~ + n3~ + n3~ n1~
[
2 2 2
+ 2 I n3~ n1~ n2~ - n3~ n2~ ) (n1~ + n2~ I)/(%4 %3 %2 %1
(n1~ - n2~ I)) ,
2 2 2 ]
(-1 + n3~ + n1~ + 2 I n1~ n2~ - n2~ ) n3~ (n1~ + n2~ I)]
2 ---------------------------------------------------------]
%4 %3 %2 %1 ]
[ 2 2 2 2
[ (n1~ + n2~ I) n3~ (-1 + n3~ + n1~ + 2 I n1~ n2~ - n2~ )
[2 ---------------------------------------------------------- ,
[ %4 %3 %2 %1 (n1~ - n2~ I)
2 2 2 2 ]
(n1~ + n2~ I) (1 + n3~ + n1~ + 2 I n1~ n2~ - n2~ )]
2 -----------------------------------------------------]
%4 %3 %2 %1 ]
%1 := n1~ + I + n2~ I + n3~ I
%2 := n1~ + I - n3~ I + n2~ I
%3 := n1~ - I + n3~ I + n2~ I
%4 := n1~ - I + n2~ I - n3~ I
% Hmm this is not going well. In the words of Dirac,
% it does not have mathematical beauty!
sqrt(ev1(1)^2+ev1(2)^2); pretty(ans)
/ 2 \1/2
| (1 + n3~) |
|-------------- + 1|
| 2 |
\(n1~ + n2~ I) /
% Is this equal to one already?
% I think I need to take a fresh approach to this. I'm not getting anywhere yet
% Signing off...
diary off
% Is this on?
diary off
|
Forum timezone: GMT+0 VF Version: 3.00b, ConfDB: Before posting please read our privacy policy. VoyForums(tm) is a Free Service from Voyager Info-Systems. Copyright © 1998-2019 Voyager Info-Systems. All Rights Reserved. |