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SandBoxHermitianIsomorphisms3

Edit detail for SandBoxHermitianIsomorphisms3 revision 2 of 11

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Editor: Bill Page Time: 2011/06/29 08:53:30 GMT-7
Note:

changed:
-s1:=solve(imag determinant ρ,ℜp3)
s1:=solve(imag determinant ρ,ℜp1)
s2:=solve(imag trace(ρ),𝔍p1)
s3:=solve(eval(imag trace(ρ*ρ),s2),ℜq2)
s4:=radicalSolve(eval(eval(imag trace(ρ*ρ*ρ),s2),s3),ℜr3)

A complex vector ℂ-space possesses many different hermitian isomorphisms $h^\dagger=h \in iso(V,V^\dagger)$ . In quantum mechanics a given operator $\rho \in End(V)$ may be said to be -hermitian if

$\rho^\dagger \circ h = h \circ \rho$

axiom

ℂ:=Complex Fraction Polynomial Integer

$\label{eq1}\hbox{\axiomType{Complex}\ } (\hbox{\axiomType{Fraction}\ } (\hbox{\axiomType{Polynomial}\ } (\hbox{\axiomType{Integer}\ })))$

(1)

Type: Type

axiom

-- dagger
htranspose(h)==map(x+->conjugate(x),transpose h)

Type: Void

axiom

)expose MCALCFN

   MultiVariableCalculusFunctions is now explicitly exposed in frame 
      initial

Theorem

The necessary conditions for an operator to possess hermitean isomorphism is that $trace ρ \in ℝ$ and $det ρ \in ℝ$ .

Two-Dimensions

axiom

p1:ℂ:=complex(ℜp1,𝔍p1)

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

(2)

Type: Complex(Fraction(Polynomial(Integer)))

axiom

q1:ℂ:=complex(ℜq1,𝔍q1)

$\label{eq3}\begin{array}{@{}l} \displaystyle � � q 1 +{ \begin{array}{@{}l} \displaystyle �� q 1 \ \cdot \ \ \displaystyle i$

(3)

Type: Complex(Fraction(Polynomial(Integer)))

axiom

p2:ℂ:=complex(ℜp2,𝔍p2)

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

(4)

Type: Complex(Fraction(Polynomial(Integer)))

axiom

q2:ℂ:=complex(ℜq2,𝔍q2)

$\label{eq5}\begin{array}{@{}l} \displaystyle � � q 2 +{ \begin{array}{@{}l} \displaystyle �� q 2 \ \cdot \ \ \displaystyle i$

(5)

Type: Complex(Fraction(Polynomial(Integer)))

axiom

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

$\label{eq6}\left[ \begin{array}{cc} {� � p 1 +{�� p 1 \ i}}&{� � q 1 +{�� q 1 \ i}} \ {� � p 2 +{�� p 2 \ i}}&{� � q 2 +{�� q 2 \ i}}$

(6)

Type: Matrix(Complex(Fraction(Polynomial(Integer))))

axiom

s1:=solve(imag determinant ρ,ℜp2)

$\label{eq7}\left[ � � p 2 ={{{� � p 1 \ �� q 2}-{� � q 1 \ �� p 2}+{� � q 2 \ �� p 1}}\over �� q 1}\right]$

(7)

Type: List(Equation(Fraction(Polynomial(Integer))))

axiom

s2:=solve(eval(imag trace ρ, s1),𝔍q2)

$\label{eq8}\left[ �� q 2 = - �� p 1 \right]$

(8)

Type: List(Equation(Fraction(Polynomial(Integer))))

axiom

eval(eval(imag trace (ρ*ρ),s1),s2)

$\label{eq9}0$

(9)

Type: Fraction(Polynomial(Integer))

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

axiom

h:Matrix ℂ:=matrix [[a,complex(b,c)],[complex(b,-c),e]]

$\label{eq10}\left[ \begin{array}{cc} a &{b +{c \ i}} \ {b -{c \ i}}& e$

(10)

Type: Matrix(Complex(Fraction(Polynomial(Integer))))

axiom

test(h = htranspose h)

axiom

Compiling function htranspose with type Matrix(Complex(Fraction(
      Polynomial(Integer)))) -> Matrix(Complex(Fraction(Polynomial(
      Integer))))

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

(11)

Type: Boolean

axiom

H:=htranspose(ρ)*h-h*ρ

$\label{eq12}\left[ \begin{array}{cc} {{\left(-{2 \ b \ �� p 2}-{2 \ a \ �� p 1}-{2 \ c \ � � p 2}\right)}\ i}&{{c \ �� q 2}+{c \ �� p 1}-{b \ � � q 2}-{a \ � � q 1}+{e \ � � p 2}+{b \ � � p 1}+{{\left(-{b \ �� q 2}-{a \ �� q 1}-{e \ �� p 2}-{b \ �� p 1}-{c \ � � q 2}+{c \ � � p 1}\right)}\ i}} \ {-{c \ �� q 2}-{c \ �� p 1}+{b \ � � q 2}+{a \ � � q 1}-{e \ � � p 2}-{b \ � � p 1}+{{\left(-{b \ �� q 2}-{a \ �� q 1}-{e \ �� p 2}-{b \ �� p 1}-{c \ � � q 2}+{c \ � � p 1}\right)}\ i}}&{{\left(-{2 \ e \ �� q 2}-{2 \ b \ �� q 1}+{2 \ c \ � � q 1}\right)}\ i}$

(12)

Type: Matrix(Complex(Fraction(Polynomial(Integer))))

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 ?)) ),
     [a,b,c,e]::List Symbol)

$\label{eq13}\left[ \begin{array}{cccc} 0 & 0 & 0 & 0 \ -{2 \ �� p 1}& -{2 \ �� p 2}& -{2 \ � � p 2}& 0 \ - � � q 1 &{- � � q 2 + � � p 1}&{�� q 2 + �� p 1}& � � p 2 \ - �� q 1 &{- �� q 2 - �� p 1}&{- � � q 2 + � � p 1}& - �� p 2 \ � � q 1 &{� � q 2 - � � p 1}&{- �� q 2 - �� p 1}& - � � p 2 \ - �� q 1 &{- �� q 2 - �� p 1}&{- � � q 2 + � � p 1}& - �� p 2 \ 0 & 0 & 0 & 0 \ 0 & -{2 \ �� q 1}&{2 \ � � q 1}& -{2 \ �� q 2}$

(13)

Type: Matrix(Fraction(Polynomial(Integer)))

The null space (kernel) of the Jacobian

axiom

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

$\label{eq14}\begin{array}{@{}l} \displaystyle \left[{ \begin{array}{@{}l} \displaystyle \left[{{- � � q 2 + � � p 1}\over �� q 1}, \right. \ \ \displaystyle \left.\: \right. \ \ \displaystyle \left.{� � q 1 \over �� q 1}, \right. \ \ \displaystyle \left.\: 1, \: 0 \right]$

(14)

Type: List(Vector(Fraction(Polynomial(Integer))))

gives the general solution to the problem.

axiom

s3:=map((x,y)+->x=y,[a,b,c,e],c*N.1+e*N.2)

$\label{eq15}\begin{array}{@{}l} \displaystyle \left[{ \begin{array}{@{}l} \displaystyle a = \ \ \displaystyle {{-{e \ �� p 2}-{c \ � � q 2}+{c \ � � p 1}}\over �� q 1}$

(15)

Type: List(Equation(Fraction(Polynomial(Integer))))

axiom

map(x+->eval(eval(eval(x,s1),s2),s3),H)

$\label{eq16}\left[ \begin{array}{cc} 0 & 0 \ 0 & 0$

(16)

Type: Matrix(Fraction(Polynomial(Complex(Integer))))

axiom

)set output tex off

axiom

)set output algebra on

Three-Dimensions

axiom

p1:ℂ:=complex(ℜp1,𝔍p1)

   (18)  ℜp1 + 𝔍p1 %i

Type: Complex(Fraction(Polynomial(Integer)))

axiom

q1:ℂ:=complex(ℜq1,𝔍q1)

   (19)  ℜq1 + 𝔍q1 %i

Type: Complex(Fraction(Polynomial(Integer)))

axiom

r1:ℂ:=complex(ℜr1,𝔍r1)

   (20)  ℜr1 + 𝔍r1 %i

Type: Complex(Fraction(Polynomial(Integer)))

axiom

p2:ℂ:=complex(ℜp2,𝔍p2)

   (21)  ℜp2 + 𝔍p2 %i

Type: Complex(Fraction(Polynomial(Integer)))

axiom

q2:ℂ:=complex(ℜq2,𝔍q2)

   (22)  ℜq2 + 𝔍q2 %i

Type: Complex(Fraction(Polynomial(Integer)))

axiom

r2:ℂ:=complex(ℜr2,𝔍r2)

   (23)  ℜr2 + 𝔍r2 %i

Type: Complex(Fraction(Polynomial(Integer)))

axiom

p3:ℂ:=complex(ℜp3,𝔍p3)

   (24)  ℜp3 + 𝔍p3 %i

Type: Complex(Fraction(Polynomial(Integer)))

axiom

q3:ℂ:=complex(ℜq3,𝔍q3)

   (25)  ℜq3 + 𝔍q3 %i

Type: Complex(Fraction(Polynomial(Integer)))

axiom

r3:ℂ:=complex(ℜr3,𝔍r3)

   (26)  ℜr3 + 𝔍r3 %i

Type: Complex(Fraction(Polynomial(Integer)))

axiom

ρ:Matrix ℂ := matrix [[p1,q1,r1],[p2,q2,r2],[p3,q3,r3]]

         +ℜp1 + 𝔍p1 %i  ℜq1 + 𝔍q1 %i  ℜr1 + 𝔍r1 %i+
         |                                        |
   (27)  |ℜp2 + 𝔍p2 %i  ℜq2 + 𝔍q2 %i  ℜr2 + 𝔍r2 %i|
         |                                        |
         +ℜp3 + 𝔍p3 %i  ℜq3 + 𝔍q3 %i  ℜr3 + 𝔍r3 %i+

Type: Matrix(Complex(Fraction(Polynomial(Integer))))

axiom

s1:=solve(imag determinant ρ,ℜp1)

   (28)
   [
     ℜp1 =
           (𝔍p1 𝔍q2 - 𝔍p2 𝔍q1 + ℜp2 ℜq1)𝔍r3 + (- 𝔍p1 𝔍q3 + 𝔍p3 𝔍q1 - ℜp3 ℜq1)𝔍r2
         + 
           (𝔍p2 𝔍q3 - 𝔍p3 𝔍q2 - ℜp2 ℜq3 + ℜp3 ℜq2)𝔍r1 - ℜp2 ℜr1 𝔍q3
         + 
           ℜp3 ℜr1 𝔍q2 + (ℜp2 ℜr3 - ℜp3 ℜr2)𝔍q1 + (- ℜq1 ℜr2 + ℜq2 ℜr1)𝔍p3
         + 
           (ℜq1 ℜr3 - ℜq3 ℜr1)𝔍p2 + (- ℜq2 ℜr3 + ℜq3 ℜr2)𝔍p1
      /
         ℜq2 𝔍r3 - ℜq3 𝔍r2 - ℜr2 𝔍q3 + ℜr3 𝔍q2
     ]

Type: List(Equation(Fraction(Polynomial(Integer))))

axiom

s2:=solve(imag trace(ρ),𝔍p1)

   (29)  [𝔍p1= - 𝔍r3 - 𝔍q2]

Type: List(Equation(Fraction(Polynomial(Integer))))

axiom

s3:=solve(eval(imag trace(ρ*ρ),s2),ℜq2)

   (30)
   [
     ℜq2 =
           (- ℜr3 + ℜp1)𝔍r3 - ℜq3 𝔍r2 - ℜp3 𝔍r1 - ℜr2 𝔍q3 + ℜp1 𝔍q2 - ℜp2 𝔍q1
         + 
           - ℜr1 𝔍p3 - ℜq1 𝔍p2
      /
         𝔍q2
     ]

Type: List(Equation(Fraction(Polynomial(Integer))))

axiom

s4:=radicalSolve(eval(eval(imag trace(ρ*ρ*ρ),s2),s3),ℜr3)

   (31)
   [
     ℜr3 =
           -
              ROOT
                         2   4
                   - 4𝔍q2 𝔍r3
                 + 
                                          3
                       4𝔍q2 𝔍q3 𝔍r2 - 8𝔍q2
                     + 
                       (- 4𝔍p2 𝔍q1 - 4ℜq3 ℜr2 + 4ℜp2 ℜq1)𝔍q2
                  *
                        3
                     𝔍r3
                 + 
                          2   2
                       ℜq3 𝔍r2
                     + 
                                2
                           (8𝔍q2  + 2ℜq3 ℜr2)𝔍q3 + (4𝔍p3 𝔍q1 - 4ℜp3 ℜq1)𝔍q2
                         + 
                           2ℜp2 ℜq3 𝔍q1 + 2ℜq1 ℜq3 𝔍p2
                      *
                         𝔍r2
                     + 
                                               2                         2   2
                       (4𝔍p2 𝔍q2 𝔍q3 - 4𝔍p3 𝔍q2  - 4ℜp2 ℜq3 𝔍q2)𝔍r1 + ℜr2 𝔍q3
                     + 
                                                                               4
                       (- 4ℜp2 ℜr1 𝔍q2 + 2ℜp2 ℜr2 𝔍q1 + 2ℜq1 ℜr2 𝔍p2)𝔍q3 - 4𝔍q2
                     + 
                                                                       2
                       (- 4𝔍p2 𝔍q1 - 8ℜq3 ℜr2 + 4ℜp3 ℜr1 + 4ℜp2 ℜq1)𝔍q2
                     + 
                       (- 4ℜp3 ℜr2 𝔍q1 - 4ℜq1 ℜr2 𝔍p3 - 4ℜq3 ℜr1 𝔍p2)𝔍q2
                     + 
                          2   2                         2   2
                       ℜp2 𝔍q1  + 2ℜp2 ℜq1 𝔍p2 𝔍q1 + ℜq1 𝔍p2
                  *
                        2
                     𝔍r3
                 + 
                           2       2
                       2ℜq3 𝔍q2 𝔍r2
                     + 
                                                   3
                           2ℜp3 ℜq3 𝔍q2 𝔍r1 + (4𝔍q2  + 4ℜq3 ℜr2 𝔍q2)𝔍q3
                         + 
                                                   2
                           (4𝔍p3 𝔍q1 - 4ℜp3 ℜq1)𝔍q2
                         + 
                           (2ℜp2 ℜq3 𝔍q1 + 2ℜq3 ℜr1 𝔍p3 + 2ℜq1 ℜq3 𝔍p2)𝔍q2
                      *
                         𝔍r2
                     + 
                                    2                              3
                           (4𝔍p2 𝔍q2  + 2ℜp3 ℜr2 𝔍q2)𝔍q3 - 4𝔍p3 𝔍q2
                         + 
                                         2
                           - 4ℜp2 ℜq3 𝔍q2  + (- 2ℜp2 ℜp3 𝔍q1 - 2ℜp3 ℜq1 𝔍p2)𝔍q2
                      *
                         𝔍r1
                     + 
                           2       2
                       2ℜr2 𝔍q2 𝔍q3
                     + 
                                         2
                           - 4ℜp2 ℜr1 𝔍q2
                         + 
                           (2ℜp2 ℜr2 𝔍q1 + 2ℜr1 ℜr2 𝔍p3 + 2ℜq1 ℜr2 𝔍p2)𝔍q2
                      *
                         𝔍q3
                     + 
                                                 3
                       (- 4ℜq3 ℜr2 + 4ℜp3 ℜr1)𝔍q2
                     + 
                                                                        2
                       (- 4ℜp3 ℜr2 𝔍q1 - 4ℜq1 ℜr2 𝔍p3 - 4ℜq3 ℜr1 𝔍p2)𝔍q2
                     + 
                       (- 2ℜp2 ℜr1 𝔍p3 𝔍q1 - 2ℜq1 ℜr1 𝔍p2 𝔍p3)𝔍q2
                  *
                     𝔍r3
                 + 
                      2   2   2
                   ℜq3 𝔍q2 𝔍r2
                 + 
                                2                  2                      2
                   (2ℜp3 ℜq3 𝔍q2 𝔍r1 + 2ℜq3 ℜr2 𝔍q2 𝔍q3 + 2ℜq3 ℜr1 𝔍p3 𝔍q2 )𝔍r2
                 + 
                      2   2   2                2                      2
                   ℜp3 𝔍q2 𝔍r1  + (2ℜp3 ℜr2 𝔍q2 𝔍q3 + 2ℜp3 ℜr1 𝔍p3 𝔍q2 )𝔍r1
                 + 
                      2   2   2                   2         2   2   2
                   ℜr2 𝔍q2 𝔍q3  + 2ℜr1 ℜr2 𝔍p3 𝔍q2 𝔍q3 + ℜr1 𝔍p3 𝔍q2
         + 
                   2
           2ℜp1 𝔍r3
         + 
               - ℜq3 𝔍r2 - 2ℜp3 𝔍r1 - ℜr2 𝔍q3 + 2ℜp1 𝔍q2 - ℜp2 𝔍q1 - 2ℜr1 𝔍p3
             + 
               - ℜq1 𝔍p2
          *
             𝔍r3
         + 
           - ℜq3 𝔍q2 𝔍r2 - ℜp3 𝔍q2 𝔍r1 - ℜr2 𝔍q2 𝔍q3 - ℜr1 𝔍p3 𝔍q2
      /
             2
         2𝔍r3  + 2𝔍q2 𝔍r3
     ,

     ℜr3 =
           ROOT
                      2   4
                - 4𝔍q2 𝔍r3
              + 
                                      3
                  (4𝔍q2 𝔍q3 𝔍r2 - 8𝔍q2  + (- 4𝔍p2 𝔍q1 - 4ℜq3 ℜr2 + 4ℜp2 ℜq1)𝔍q2)
               *
                     3
                  𝔍r3
              + 
                       2   2
                    ℜq3 𝔍r2
                  + 
                             2
                        (8𝔍q2  + 2ℜq3 ℜr2)𝔍q3 + (4𝔍p3 𝔍q1 - 4ℜp3 ℜq1)𝔍q2
                      + 
                        2ℜp2 ℜq3 𝔍q1 + 2ℜq1 ℜq3 𝔍p2
                   *
                      𝔍r2
                  + 
                                            2                         2   2
                    (4𝔍p2 𝔍q2 𝔍q3 - 4𝔍p3 𝔍q2  - 4ℜp2 ℜq3 𝔍q2)𝔍r1 + ℜr2 𝔍q3
                  + 
                                                                            4
                    (- 4ℜp2 ℜr1 𝔍q2 + 2ℜp2 ℜr2 𝔍q1 + 2ℜq1 ℜr2 𝔍p2)𝔍q3 - 4𝔍q2
                  + 
                                                                    2
                    (- 4𝔍p2 𝔍q1 - 8ℜq3 ℜr2 + 4ℜp3 ℜr1 + 4ℜp2 ℜq1)𝔍q2
                  + 
                                                                           2   2
                    (- 4ℜp3 ℜr2 𝔍q1 - 4ℜq1 ℜr2 𝔍p3 - 4ℜq3 ℜr1 𝔍p2)𝔍q2 + ℜp2 𝔍q1
                  + 
                                          2   2
                    2ℜp2 ℜq1 𝔍p2 𝔍q1 + ℜq1 𝔍p2
               *
                     2
                  𝔍r3
              + 
                        2       2
                    2ℜq3 𝔍q2 𝔍r2
                  + 
                                                3
                        2ℜp3 ℜq3 𝔍q2 𝔍r1 + (4𝔍q2  + 4ℜq3 ℜr2 𝔍q2)𝔍q3
                      + 
                                                2
                        (4𝔍p3 𝔍q1 - 4ℜp3 ℜq1)𝔍q2
                      + 
                        (2ℜp2 ℜq3 𝔍q1 + 2ℜq3 ℜr1 𝔍p3 + 2ℜq1 ℜq3 𝔍p2)𝔍q2
                   *
                      𝔍r2
                  + 
                                 2                              3
                        (4𝔍p2 𝔍q2  + 2ℜp3 ℜr2 𝔍q2)𝔍q3 - 4𝔍p3 𝔍q2
                      + 
                                      2
                        - 4ℜp2 ℜq3 𝔍q2  + (- 2ℜp2 ℜp3 𝔍q1 - 2ℜp3 ℜq1 𝔍p2)𝔍q2
                   *
                      𝔍r1
                  + 
                        2       2
                    2ℜr2 𝔍q2 𝔍q3
                  + 
                                      2
                        - 4ℜp2 ℜr1 𝔍q2
                      + 
                        (2ℜp2 ℜr2 𝔍q1 + 2ℜr1 ℜr2 𝔍p3 + 2ℜq1 ℜr2 𝔍p2)𝔍q2
                   *
                      𝔍q3
                  + 
                                              3
                    (- 4ℜq3 ℜr2 + 4ℜp3 ℜr1)𝔍q2
                  + 
                                                                     2
                    (- 4ℜp3 ℜr2 𝔍q1 - 4ℜq1 ℜr2 𝔍p3 - 4ℜq3 ℜr1 𝔍p2)𝔍q2
                  + 
                    (- 2ℜp2 ℜr1 𝔍p3 𝔍q1 - 2ℜq1 ℜr1 𝔍p2 𝔍p3)𝔍q2
               *
                  𝔍r3
              + 
                   2   2   2
                ℜq3 𝔍q2 𝔍r2
              + 
                             2                  2                      2
                (2ℜp3 ℜq3 𝔍q2 𝔍r1 + 2ℜq3 ℜr2 𝔍q2 𝔍q3 + 2ℜq3 ℜr1 𝔍p3 𝔍q2 )𝔍r2
              + 
                   2   2   2                2                      2
                ℜp3 𝔍q2 𝔍r1  + (2ℜp3 ℜr2 𝔍q2 𝔍q3 + 2ℜp3 ℜr1 𝔍p3 𝔍q2 )𝔍r1
              + 
                   2   2   2                   2         2   2   2
                ℜr2 𝔍q2 𝔍q3  + 2ℜr1 ℜr2 𝔍p3 𝔍q2 𝔍q3 + ℜr1 𝔍p3 𝔍q2
         + 
                   2
           2ℜp1 𝔍r3
         + 
               - ℜq3 𝔍r2 - 2ℜp3 𝔍r1 - ℜr2 𝔍q3 + 2ℜp1 𝔍q2 - ℜp2 𝔍q1 - 2ℜr1 𝔍p3
             + 
               - ℜq1 𝔍p2
          *
             𝔍r3
         + 
           - ℜq3 𝔍q2 𝔍r2 - ℜp3 𝔍q2 𝔍r1 - ℜr2 𝔍q2 𝔍q3 - ℜr1 𝔍p3 𝔍q2
      /
             2
         2𝔍r3  + 2𝔍q2 𝔍r3
     ]

Type: List(Equation(Expression(Integer)))

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(ℜc,𝔍c)], _
                    [complex(ℜb,-𝔍b),ℜe,             complex(ℜd,𝔍d)], _
                    [complex(ℜc,-𝔍c),complex(ℜd,-𝔍d),ℜf            ]]

         +    ℜa      ℜb + 𝔍b %i  ℜc + 𝔍c %i+
         |                                  |
   (32)  |ℜb - 𝔍b %i      ℜe      ℜd + 𝔍d %i|
         |                                  |
         +ℜc - 𝔍c %i  ℜd - 𝔍d %i      ℜf    +

Type: Matrix(Complex(Fraction(Polynomial(Integer))))

axiom

test(h = htranspose h)

   (33)  true

Type: Boolean

axiom

H:=htranspose(ρ)*h-h*ρ

   (34)
   [
     [(- 2ℜc 𝔍p3 - 2ℜb 𝔍p2 - 2ℜa 𝔍p1 - 2ℜp3 𝔍c - 2ℜp2 𝔍b)%i,

         𝔍c 𝔍q3 + 𝔍b 𝔍q2 - 𝔍d 𝔍p3 + 𝔍b 𝔍p1 - ℜc ℜq3 - ℜb ℜq2 - ℜa ℜq1 + ℜd ℜp3
       + 
         ℜe ℜp2 + ℜb ℜp1
       + 
             - ℜc 𝔍q3 - ℜb 𝔍q2 - ℜa 𝔍q1 - ℜd 𝔍p3 - ℜe 𝔍p2 - ℜb 𝔍p1 - ℜp3 𝔍d
           + 
             - ℜq3 𝔍c + (- ℜq2 + ℜp1)𝔍b
        *
           %i
       ,

         𝔍c 𝔍r3 + 𝔍b 𝔍r2 + 𝔍d 𝔍p2 + 𝔍c 𝔍p1 - ℜc ℜr3 - ℜb ℜr2 - ℜa ℜr1 + ℜf ℜp3
       + 
         ℜd ℜp2 + ℜc ℜp1
       + 
             - ℜc 𝔍r3 - ℜb 𝔍r2 - ℜa 𝔍r1 - ℜf 𝔍p3 - ℜd 𝔍p2 - ℜc 𝔍p1 + ℜp2 𝔍d
           + 
             (- ℜr3 + ℜp1)𝔍c - ℜr2 𝔍b
        *
           %i
       ]
     ,

     [
         - 𝔍c 𝔍q3 - 𝔍b 𝔍q2 + 𝔍d 𝔍p3 - 𝔍b 𝔍p1 + ℜc ℜq3 + ℜb ℜq2 + ℜa ℜq1 - ℜd ℜp3
       + 
         - ℜe ℜp2 - ℜb ℜp1
       + 
             - ℜc 𝔍q3 - ℜb 𝔍q2 - ℜa 𝔍q1 - ℜd 𝔍p3 - ℜe 𝔍p2 - ℜb 𝔍p1 - ℜp3 𝔍d
           + 
             - ℜq3 𝔍c + (- ℜq2 + ℜp1)𝔍b
        *
           %i
       ,
      (- 2ℜd 𝔍q3 - 2ℜe 𝔍q2 - 2ℜb 𝔍q1 - 2ℜq3 𝔍d + 2ℜq1 𝔍b)%i,

         𝔍d 𝔍r3 - 𝔍b 𝔍r1 + 𝔍d 𝔍q2 + 𝔍c 𝔍q1 - ℜd ℜr3 - ℜe ℜr2 - ℜb ℜr1 + ℜf ℜq3
       + 
         ℜd ℜq2 + ℜc ℜq1
       + 
             - ℜd 𝔍r3 - ℜe 𝔍r2 - ℜb 𝔍r1 - ℜf 𝔍q3 - ℜd 𝔍q2 - ℜc 𝔍q1
           + 
             (- ℜr3 + ℜq2)𝔍d + ℜq1 𝔍c + ℜr1 𝔍b
        *
           %i
       ]
     ,

     [
         - 𝔍c 𝔍r3 - 𝔍b 𝔍r2 - 𝔍d 𝔍p2 - 𝔍c 𝔍p1 + ℜc ℜr3 + ℜb ℜr2 + ℜa ℜr1 - ℜf ℜp3
       + 
         - ℜd ℜp2 - ℜc ℜp1
       + 
             - ℜc 𝔍r3 - ℜb 𝔍r2 - ℜa 𝔍r1 - ℜf 𝔍p3 - ℜd 𝔍p2 - ℜc 𝔍p1 + ℜp2 𝔍d
           + 
             (- ℜr3 + ℜp1)𝔍c - ℜr2 𝔍b
        *
           %i
       ,

         - 𝔍d 𝔍r3 + 𝔍b 𝔍r1 - 𝔍d 𝔍q2 - 𝔍c 𝔍q1 + ℜd ℜr3 + ℜe ℜr2 + ℜb ℜr1 - ℜf ℜq3
       + 
         - ℜd ℜq2 - ℜc ℜq1
       + 
             - ℜd 𝔍r3 - ℜe 𝔍r2 - ℜb 𝔍r1 - ℜf 𝔍q3 - ℜd 𝔍q2 - ℜc 𝔍q1
           + 
             (- ℜr3 + ℜq2)𝔍d + ℜq1 𝔍c + ℜr1 𝔍b
        *
           %i
       ,
      (- 2ℜf 𝔍r3 - 2ℜd 𝔍r2 - 2ℜc 𝔍r1 + 2ℜr2 𝔍d + 2ℜr1 𝔍c)%i]
     ]

Type: Matrix(Complex(Fraction(Polynomial(Integer))))

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

K:=concat( map(x+->[real x, imag x], concat(H::List List ?)))::List Polynomial Integer

   (35)
   [0, - 2ℜc 𝔍p3 - 2ℜb 𝔍p2 - 2ℜa 𝔍p1 - 2ℜp3 𝔍c - 2ℜp2 𝔍b,

       𝔍c 𝔍q3 + 𝔍b 𝔍q2 - 𝔍d 𝔍p3 + 𝔍b 𝔍p1 - ℜc ℜq3 - ℜb ℜq2 - ℜa ℜq1 + ℜd ℜp3
     + 
       ℜe ℜp2 + ℜb ℜp1
     ,

       - ℜc 𝔍q3 - ℜb 𝔍q2 - ℜa 𝔍q1 - ℜd 𝔍p3 - ℜe 𝔍p2 - ℜb 𝔍p1 - ℜp3 𝔍d - ℜq3 𝔍c
     + 
       (- ℜq2 + ℜp1)𝔍b
     ,

       𝔍c 𝔍r3 + 𝔍b 𝔍r2 + 𝔍d 𝔍p2 + 𝔍c 𝔍p1 - ℜc ℜr3 - ℜb ℜr2 - ℜa ℜr1 + ℜf ℜp3
     + 
       ℜd ℜp2 + ℜc ℜp1
     ,

       - ℜc 𝔍r3 - ℜb 𝔍r2 - ℜa 𝔍r1 - ℜf 𝔍p3 - ℜd 𝔍p2 - ℜc 𝔍p1 + ℜp2 𝔍d
     + 
       (- ℜr3 + ℜp1)𝔍c - ℜr2 𝔍b
     ,

       - 𝔍c 𝔍q3 - 𝔍b 𝔍q2 + 𝔍d 𝔍p3 - 𝔍b 𝔍p1 + ℜc ℜq3 + ℜb ℜq2 + ℜa ℜq1 - ℜd ℜp3
     + 
       - ℜe ℜp2 - ℜb ℜp1
     ,

       - ℜc 𝔍q3 - ℜb 𝔍q2 - ℜa 𝔍q1 - ℜd 𝔍p3 - ℜe 𝔍p2 - ℜb 𝔍p1 - ℜp3 𝔍d - ℜq3 𝔍c
     + 
       (- ℜq2 + ℜp1)𝔍b
     ,
    0, - 2ℜd 𝔍q3 - 2ℜe 𝔍q2 - 2ℜb 𝔍q1 - 2ℜq3 𝔍d + 2ℜq1 𝔍b,

       𝔍d 𝔍r3 - 𝔍b 𝔍r1 + 𝔍d 𝔍q2 + 𝔍c 𝔍q1 - ℜd ℜr3 - ℜe ℜr2 - ℜb ℜr1 + ℜf ℜq3
     + 
       ℜd ℜq2 + ℜc ℜq1
     ,

       - ℜd 𝔍r3 - ℜe 𝔍r2 - ℜb 𝔍r1 - ℜf 𝔍q3 - ℜd 𝔍q2 - ℜc 𝔍q1 + (- ℜr3 + ℜq2)𝔍d
     + 
       ℜq1 𝔍c + ℜr1 𝔍b
     ,

       - 𝔍c 𝔍r3 - 𝔍b 𝔍r2 - 𝔍d 𝔍p2 - 𝔍c 𝔍p1 + ℜc ℜr3 + ℜb ℜr2 + ℜa ℜr1 - ℜf ℜp3
     + 
       - ℜd ℜp2 - ℜc ℜp1
     ,

       - ℜc 𝔍r3 - ℜb 𝔍r2 - ℜa 𝔍r1 - ℜf 𝔍p3 - ℜd 𝔍p2 - ℜc 𝔍p1 + ℜp2 𝔍d
     + 
       (- ℜr3 + ℜp1)𝔍c - ℜr2 𝔍b
     ,

       - 𝔍d 𝔍r3 + 𝔍b 𝔍r1 - 𝔍d 𝔍q2 - 𝔍c 𝔍q1 + ℜd ℜr3 + ℜe ℜr2 + ℜb ℜr1 - ℜf ℜq3
     + 
       - ℜd ℜq2 - ℜc ℜq1
     ,

       - ℜd 𝔍r3 - ℜe 𝔍r2 - ℜb 𝔍r1 - ℜf 𝔍q3 - ℜd 𝔍q2 - ℜc 𝔍q1 + (- ℜr3 + ℜq2)𝔍d
     + 
       ℜq1 𝔍c + ℜr1 𝔍b
     ,
    0, - 2ℜf 𝔍r3 - 2ℜd 𝔍r2 - 2ℜc 𝔍r1 + 2ℜr2 𝔍d + 2ℜr1 𝔍c]

Type: List(Polynomial(Integer))

axiom

)set output tex off

axiom

)set output algebra on

expr(K::InputForm)

   (36)
   [0, - 2ℜc 𝔍p3 - 2ℜb 𝔍p2 - 2ℜa 𝔍p1 - 2ℜp3 𝔍c - 2ℜp2 𝔍b,

       𝔍c 𝔍q3 + 𝔍b 𝔍q2 - 𝔍d 𝔍p3 + 𝔍b 𝔍p1 - ℜc ℜq3 - ℜb ℜq2 - ℜa ℜq1 + ℜd ℜp3
     + 
       ℜe ℜp2 + ℜb ℜp1
     ,

       - ℜc 𝔍q3 - ℜb 𝔍q2 - ℜa 𝔍q1 - ℜd 𝔍p3 - ℜe 𝔍p2 - ℜb 𝔍p1 - ℜp3 𝔍d - ℜq3 𝔍c
     + 
       (- ℜq2 + ℜp1)𝔍b
     ,

       𝔍c 𝔍r3 + 𝔍b 𝔍r2 + 𝔍d 𝔍p2 + 𝔍c 𝔍p1 - ℜc ℜr3 - ℜb ℜr2 - ℜa ℜr1 + ℜf ℜp3
     + 
       ℜd ℜp2 + ℜc ℜp1
     ,

       - ℜc 𝔍r3 - ℜb 𝔍r2 - ℜa 𝔍r1 - ℜf 𝔍p3 - ℜd 𝔍p2 - ℜc 𝔍p1 + ℜp2 𝔍d
     + 
       (- ℜr3 + ℜp1)𝔍c - ℜr2 𝔍b
     ,

       - 𝔍c 𝔍q3 - 𝔍b 𝔍q2 + 𝔍d 𝔍p3 - 𝔍b 𝔍p1 + ℜc ℜq3 + ℜb ℜq2 + ℜa ℜq1 - ℜd ℜp3
     + 
       - ℜe ℜp2 - ℜb ℜp1
     ,

       - ℜc 𝔍q3 - ℜb 𝔍q2 - ℜa 𝔍q1 - ℜd 𝔍p3 - ℜe 𝔍p2 - ℜb 𝔍p1 - ℜp3 𝔍d - ℜq3 𝔍c
     + 
       (- ℜq2 + ℜp1)𝔍b
     ,
    0, - 2ℜd 𝔍q3 - 2ℜe 𝔍q2 - 2ℜb 𝔍q1 - 2ℜq3 𝔍d + 2ℜq1 𝔍b,

       𝔍d 𝔍r3 - 𝔍b 𝔍r1 + 𝔍d 𝔍q2 + 𝔍c 𝔍q1 - ℜd ℜr3 - ℜe ℜr2 - ℜb ℜr1 + ℜf ℜq3
     + 
       ℜd ℜq2 + ℜc ℜq1
     ,

       - ℜd 𝔍r3 - ℜe 𝔍r2 - ℜb 𝔍r1 - ℜf 𝔍q3 - ℜd 𝔍q2 - ℜc 𝔍q1 + (- ℜr3 + ℜq2)𝔍d
     + 
       ℜq1 𝔍c + ℜr1 𝔍b
     ,

       - 𝔍c 𝔍r3 - 𝔍b 𝔍r2 - 𝔍d 𝔍p2 - 𝔍c 𝔍p1 + ℜc ℜr3 + ℜb ℜr2 + ℜa ℜr1 - ℜf ℜp3
     + 
       - ℜd ℜp2 - ℜc ℜp1
     ,

       - ℜc 𝔍r3 - ℜb 𝔍r2 - ℜa 𝔍r1 - ℜf 𝔍p3 - ℜd 𝔍p2 - ℜc 𝔍p1 + ℜp2 𝔍d
     + 
       (- ℜr3 + ℜp1)𝔍c - ℜr2 𝔍b
     ,

       - 𝔍d 𝔍r3 + 𝔍b 𝔍r1 - 𝔍d 𝔍q2 - 𝔍c 𝔍q1 + ℜd ℜr3 + ℜe ℜr2 + ℜb ℜr1 - ℜf ℜq3
     + 
       - ℜd ℜq2 - ℜc ℜq1
     ,

       - ℜd 𝔍r3 - ℜe 𝔍r2 - ℜb 𝔍r1 - ℜf 𝔍q3 - ℜd 𝔍q2 - ℜc 𝔍q1 + (- ℜr3 + ℜq2)𝔍d
     + 
       ℜq1 𝔍c + ℜr1 𝔍b
     ,
    0, - 2ℜf 𝔍r3 - 2ℜd 𝔍r2 - 2ℜc 𝔍r1 + 2ℜr2 𝔍d + 2ℜr1 𝔍c]

Type: OutputForm?

The null space (kernel) of the Jacobian Axiom output parse error!