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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 V 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 h-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 ρ to possess hermitean isomorphism h 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 H=0 for unknown manifold of hermitian isomorphisms h.

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 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.

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 H=0 for unknown manifold of hermitian isomorphisms h.

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 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.

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!