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SandBoxHermitianIsomorphisms4

Edit detail for SandBoxHermitianIsomorphisms4 revision 1 of 2

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Editor: Bill Page Time: 2011/07/03 17:36:16 GMT-7
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EigenVectors and Diagonalization

\begin{axiom}
p1:ℂ:=complex(ℜp1,𝔍p1)
q1:ℂ:=complex(ℜq1,𝔍q1)
p2:ℂ:=complex(ℜp2,𝔍p2)
q2:ℂ:=complex(ℜq2,𝔍q2)
ρ:Matrix ℂ := matrix [[p1,q1],[p2,q2]]
\end{axiom}
\begin{axiom}
s1:=solve(imag determinant ρ,ℜp2)
s2:=solve(eval(imag trace ρ,s1),𝔍p1)
s3:=solve(eval(eval(imag trace(ρ*ρ),s1), s2),ℜp1)
eval(eval(imag trace (ρ*ρ),s1),s2)
\end{axiom}
\begin{axiom}
C:=eval(eval(characteristicPolynomial ρ,s1),s2)
C0:=zerosOf(C)
#C0
imag(C0.1)
imag(C0.2)
\end{axiom}

Given an operator $ρ \in End V$, one must find the tensor $H=0$
for unknown manifold of hermitian isomorphisms $h$.
\begin{axiom}
h:Matrix ℂ:=matrix [[ℜa,complex(ℜb,𝔍b)],[complex(ℜb,-𝔍b),ℜe]]
test(h = htranspose h)
H:=htranspose(ρ)*h-h*ρ
\end{axiom}

We wish to find expressions for $h$ in terms of the components of
$ρ$. To do this we will determine how the components of $H$ depend
on the components of $h$.
\begin{axiom}
J:=jacobian(concat( map(x+->[real x, imag x], concat(H::List List ?)) ),
     [ℜa,ℜb,𝔍b,ℜe]::List Symbol)
\end{axiom}
The null space (kernel) of the Jacobian
\begin{axiom}
N:=nullSpace(map(x+->eval(eval(x,s1),s2),J))
\end{axiom}
gives the general solution to the problem.
\begin{axiom}
s4:=map((x,y)+->x=y,[ℜa,ℜb,𝔍b,ℜe],𝔍b*N.1+ℜe*N.2)
H0:=map(x+->eval(eval(eval(x,s1),s2),s4),H)
h0:=map(x+->eval(eval(eval(x,s1),s2),s4),h)
\end{axiom}

\begin{axiom}
ρ0:=map(x+->eval(eval(x,s1),s2),ρ)
E:=eigenvalues(ρ0)
E0:=eigenvector(E.1,ρ0)
E1:=map(x+->eval(x,%D=C0.1),E0.1)
E2:=map(x+->eval(x,%D=C0.2),E0.1)
test(ρ0*E1=C0(1)*E1)
test(ρ0*E2=C0(2)*E2)
EE := horizConcat(E1,E2)
EI := inverse EE
ρ1:=EI*ρ0*EE
h1:=map(x+->eval(eval(x,s1),s2),h)
hh:=EI*h1*EE
htranspose(h)==map(x+->conjugate(x::Complex Expression Integer),transpose h)
H1:=htranspose(ρ1)*hh-hh*ρ1
J1:=jacobian(concat( map(x+->[real x, imag x], concat(H1::List List ?)) ),
     [ℜa,ℜb,𝔍b,ℜe]::List Symbol)
N1:=nullSpace(map(x+->eval(eval(x,s1),s2),J1))
s5:=map((x,y)+->x=y,[ℜa,ℜb,𝔍b,ℜe],t*N1.1)
map(x+->eval(x,s5),H1)
\end{axiom}

\begin{axiom}
h2:=EE*map(x+->eval(x,s5),hh)*EI
H0:=map(x+->eval(eval(eval(x,s1),s2),s5),H)
\end{axiom}

EigenVectors? and Diagonalization

axiom

p1:ℂ:=complex(ℜp1,𝔍p1)

$\label{eq1}\begin{array}{@{}l} \displaystyle � � p 1 +{ \begin{array}{@{}l} \displaystyle �� p 1 \ \cdot \ \ \displaystyle i$

(1)

Type: Complex(Polynomial(Integer))

axiom

q1:ℂ:=complex(ℜq1,𝔍q1)

$\label{eq2}\begin{array}{@{}l} \displaystyle � � q 1 +{ \begin{array}{@{}l} \displaystyle �� q 1 \ \cdot \ \ \displaystyle i$

(2)

Type: Complex(Polynomial(Integer))

axiom

p2:ℂ:=complex(ℜp2,𝔍p2)

$\label{eq3}\begin{array}{@{}l} \displaystyle � � p 2 +{ \begin{array}{@{}l} \displaystyle �� p 2 \ \cdot \ \ \displaystyle i$

(3)

Type: Complex(Polynomial(Integer))

axiom

q2:ℂ:=complex(ℜq2,𝔍q2)

$\label{eq4}\begin{array}{@{}l} \displaystyle � � q 2 +{ \begin{array}{@{}l} \displaystyle �� q 2 \ \cdot \ \ \displaystyle i$

(4)

Type: Complex(Polynomial(Integer))

axiom

ρ:Matrix ℂ := matrix [[p1,q1],[p2,q2]]

   Matrix(ℂ) is not a valid type.

axiom

s1:=solve(imag determinant ρ,ℜp2)

   There are 3 exposed and 1 unexposed library operations named 
      determinant having 1 argument(s) but none was determined to be 
      applicable. Use HyperDoc Browse, or issue
                           )display op determinant
      to learn more about the available operations. Perhaps 
      package-calling the operation or using coercions on the arguments
      will allow you to apply the operation.

   Cannot find a definition or applicable library operation named 
      determinant with argument type(s) 
                                 Variable(ρ)

      Perhaps you should use "@" to indicate the required return type, 
      or "$" to specify which version of the function you need.
s2:=solve(eval(imag trace ρ,s1),𝔍p1)

   There are 3 exposed and 0 unexposed library operations named trace 
      having 1 argument(s) but none was determined to be applicable. 
      Use HyperDoc Browse, or issue
                              )display op trace
      to learn more about the available operations. Perhaps 
      package-calling the operation or using coercions on the arguments
      will allow you to apply the operation.

   Cannot find a definition or applicable library operation named trace
      with argument type(s) 
                                 Variable(ρ)

      Perhaps you should use "@" to indicate the required return type, 
      or "$" to specify which version of the function you need.
s3:=solve(eval(eval(imag trace(ρ*ρ),s1), s2),ℜp1)

   There are 3 exposed and 0 unexposed library operations named trace 
      having 1 argument(s) but none was determined to be applicable. 
      Use HyperDoc Browse, or issue
                              )display op trace
      to learn more about the available operations. Perhaps 
      package-calling the operation or using coercions on the arguments
      will allow you to apply the operation.

   Cannot find a definition or applicable library operation named trace
      with argument type(s) 
                             Polynomial(Integer)

      Perhaps you should use "@" to indicate the required return type, 
      or "$" to specify which version of the function you need.
eval(eval(imag trace (ρ*ρ),s1),s2)

   There are 3 exposed and 0 unexposed library operations named trace 
      having 1 argument(s) but none was determined to be applicable. 
      Use HyperDoc Browse, or issue
                              )display op trace
      to learn more about the available operations. Perhaps 
      package-calling the operation or using coercions on the arguments
      will allow you to apply the operation.

   Cannot find a definition or applicable library operation named trace
      with argument type(s) 
                             Polynomial(Integer)

      Perhaps you should use "@" to indicate the required return type, 
      or "$" to specify which version of the function you need.

axiom

C:=eval(eval(characteristicPolynomial ρ,s1),s2)

   There are 4 exposed and 1 unexposed library operations named 
      characteristicPolynomial having 1 argument(s) but none was 
      determined to be applicable. Use HyperDoc Browse, or issue
                    )display op characteristicPolynomial
      to learn more about the available operations. Perhaps 
      package-calling the operation or using coercions on the arguments
      will allow you to apply the operation.

   Cannot find a definition or applicable library operation named 
      characteristicPolynomial with argument type(s) 
                                 Variable(ρ)

      Perhaps you should use "@" to indicate the required return type, 
      or "$" to specify which version of the function you need.
C0:=zerosOf(C)

$\label{eq5}\left[ 0 \right]$

(5)

Type: List(AlgebraicNumber?)

axiom

#C0

$\label{eq6}1$

(6)

Type: PositiveInteger?

axiom

imag(C0.1)

$\label{eq7}0$

(7)

Type: Expression(Integer)

axiom

imag(C0.2)

   >> Error detected within library code:
   index out of range

Given an operator $ρ \in End V$ , one must find the tensor for unknown manifold of hermitian isomorphisms .

axiom

h:Matrix ℂ:=matrix [[ℜa,complex(ℜb,𝔍b)],[complex(ℜb,-𝔍b),ℜe]]

   Matrix(ℂ) is not a valid type.
test(h = htranspose h)

   There are no library operations named htranspose 
      Use HyperDoc Browse or issue
                             )what op htranspose
      to learn if there is any operation containing " htranspose " in 
      its name.

   Cannot find a definition or applicable library operation named 
      htranspose with argument type(s) 
                                 Variable(h)

      Perhaps you should use "@" to indicate the required return type, 
      or "$" to specify which version of the function you need.
H:=htranspose(ρ)*h-h*ρ

   There are no library operations named htranspose 
      Use HyperDoc Browse or issue
                             )what op htranspose
      to learn if there is any operation containing " htranspose " in 
      its name.

   Cannot find a definition or applicable library operation named 
      htranspose with argument type(s) 
                                 Variable(ρ)

      Perhaps you should use "@" to indicate the required return type, 
      or "$" to specify which version of the function you need.

We wish to find expressions for in terms of the components of . To do this we will determine how the components of depend on the components of .

axiom

J:=jacobian(concat( map(x+->[real x, imag x], concat(H::List List ?)) ),
     [ℜa,ℜb,𝔍b,ℜe]::List Symbol)

   Cannot convert from type Variable(H) to List(List(?)) for value
   H

The null space (kernel) of the Jacobian

axiom

N:=nullSpace(map(x+->eval(eval(x,s1),s2),J))

   There are 3 exposed and 3 unexposed library operations named 
      nullSpace having 1 argument(s) but none was determined to be 
      applicable. Use HyperDoc Browse, or issue
                            )display op nullSpace
      to learn more about the available operations. Perhaps 
      package-calling the operation or using coercions on the arguments
      will allow you to apply the operation.

   Cannot find a definition or applicable library operation named 
      nullSpace with argument type(s) 
                             Polynomial(Integer)

      Perhaps you should use "@" to indicate the required return type, 
      or "$" to specify which version of the function you need.

gives the general solution to the problem.

axiom

s4:=map((x,y)+->x=y,[ℜa,ℜb,𝔍b,ℜe],𝔍b*N.1+ℜe*N.2)

   There are no library operations named N 
      Use HyperDoc Browse or issue
                                 )what op N
      to learn if there is any operation containing " N " in its name.

   Cannot find a definition or applicable library operation named N 
      with argument type(s) 
                               PositiveInteger

      Perhaps you should use "@" to indicate the required return type, 
      or "$" to specify which version of the function you need.
H0:=map(x+->eval(eval(eval(x,s1),s2),s4),H)

$\label{eq8}H$

(8)

Type: Polynomial(Integer)

axiom

h0:=map(x+->eval(eval(eval(x,s1),s2),s4),h)

$\label{eq9}h$

(9)

Type: Polynomial(Integer)

axiom

ρ0:=map(x+->eval(eval(x,s1),s2),ρ)

$\label{eq10}��$

(10)

Type: Polynomial(Integer)

axiom

E:=eigenvalues(ρ0)

   There are 1 exposed and 0 unexposed library operations named 
      eigenvalues having 1 argument(s) but none was determined to be 
      applicable. Use HyperDoc Browse, or issue
                           )display op eigenvalues
      to learn more about the available operations. Perhaps 
      package-calling the operation or using coercions on the arguments
      will allow you to apply the operation.

   Cannot find a definition or applicable library operation named 
      eigenvalues with argument type(s) 
                             Polynomial(Integer)

      Perhaps you should use "@" to indicate the required return type, 
      or "$" to specify which version of the function you need.
E0:=eigenvector(E.1,ρ0)

   There are no library operations named E 
      Use HyperDoc Browse or issue
                                 )what op E
      to learn if there is any operation containing " E " in its name.

   Cannot find a definition or applicable library operation named E 
      with argument type(s) 
                               PositiveInteger

      Perhaps you should use "@" to indicate the required return type, 
      or "$" to specify which version of the function you need.
E1:=map(x+->eval(x,%D=C0.1),E0.1)

   There are no library operations named E0 
      Use HyperDoc Browse or issue
                                 )what op E0
      to learn if there is any operation containing " E0 " in its name.

   Cannot find a definition or applicable library operation named E0 
      with argument type(s) 
                               PositiveInteger

      Perhaps you should use "@" to indicate the required return type, 
      or "$" to specify which version of the function you need.
E2:=map(x+->eval(x,%D=C0.2),E0.1)

   There are no library operations named E0 
      Use HyperDoc Browse or issue
                                 )what op E0
      to learn if there is any operation containing " E0 " in its name.

   Cannot find a definition or applicable library operation named E0 
      with argument type(s) 
                               PositiveInteger

      Perhaps you should use "@" to indicate the required return type, 
      or "$" to specify which version of the function you need.
test(ρ0*E1=C0(1)*E1)

$\label{eq11} \mbox{\rm false}$

(11)

Type: Boolean

axiom

test(ρ0*E2=C0(2)*E2)

   >> Error detected within library code:
   index out of range

axiom

h2:=EE*map(x+->eval(x,s5),hh)*EI

$\label{eq12}EE \ EI \ hh$

(12)

Type: Polynomial(Integer)

axiom

H0:=map(x+->eval(eval(eval(x,s1),s2),s5),H)

$\label{eq13}H$

(13)

Type: Polynomial(Integer)