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A complex vector ℂ-space possesses many different hermitian isomorphisms . In quantum mechanics a given operator may be said to be -hermitian if

axiom
)set output tex off

axiom
)set output algebra on

axiom
ℂ:=Complex Fraction Polynomial Integer
(1)  Complex(Fraction(Polynomial(Integer)))
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 and .

Two-Dimensions

axiom
p1:ℂ:=complex(ℜp1,𝔍p1)
(3)  ℜp1 + 𝔍p1 %i
Type: Complex(Fraction(Polynomial(Integer)))
axiom
q1:ℂ:=complex(ℜq1,𝔍q1)
(4)  ℜq1 + 𝔍q1 %i
Type: Complex(Fraction(Polynomial(Integer)))
axiom
p2:ℂ:=complex(ℜp2,𝔍p2)
(5)  ℜp2 + 𝔍p2 %i
Type: Complex(Fraction(Polynomial(Integer)))
axiom
q2:ℂ:=complex(ℜq2,𝔍q2)
(6)  ℜq2 + 𝔍q2 %i
Type: Complex(Fraction(Polynomial(Integer)))
axiom
ρ:Matrix ℂ := matrix [[p1,q1],[p2,q2]]
+ℜp1 + 𝔍p1 %i  ℜq1 + 𝔍q1 %i+
(7)  |                          |
+ℜp2 + 𝔍p2 %i  ℜq2 + 𝔍q2 %i+
Type: Matrix(Complex(Fraction(Polynomial(Integer))))

axiom
s1:=solve(imag determinant ρ,ℜp2)
ℜp1 𝔍q2 - ℜq1 𝔍p2 + ℜq2 𝔍p1
(8)  [ℜp2= ---------------------------]
𝔍q1
Type: List(Equation(Fraction(Polynomial(Integer))))
axiom
s2:=solve(eval(imag trace ρ,s1),𝔍p1)
(9)  [𝔍p1= - 𝔍q2]
Type: List(Equation(Fraction(Polynomial(Integer))))
axiom
s3:=solve(eval(eval(imag trace(ρ*ρ),s1), s2),ℜp1)
(10)  [0= 0]
Type: List(Equation(Fraction(Polynomial(Integer))))
axiom
eval(eval(imag trace (ρ*ρ),s1),s2)
(11)  0
Type: Fraction(Polynomial(Integer))

axiom
C:=eval(eval(characteristicPolynomial ρ,s1),s2)
(12)
2                                   2
𝔍q1 𝔍q2  + (ℜq1 ℜq2 - ℜp1 ℜq1)𝔍q2 + 𝔍p2 𝔍q1
+
2          2
((ℜp1 - %A)ℜq2 - %A ℜp1 + %A )𝔍q1 + ℜq1 𝔍p2
/
𝔍q1
Type: Fraction(Polynomial(Complex(Integer)))
axiom
C0:=zerosOf(C)
(13)
[
ROOT
2                                        2
- 4𝔍q1 𝔍q2  + (- 4ℜq1 ℜq2 + 4ℜp1 ℜq1)𝔍q2 - 4𝔍p2 𝔍q1
+
2                 2           2
(ℜq2  - 2ℜp1 ℜq2 + ℜp1 )𝔍q1 - 4ℜq1 𝔍p2
/
𝔍q1
+
ℜq2 + ℜp1
/
2
,
-
ROOT
2                                        2
- 4𝔍q1 𝔍q2  + (- 4ℜq1 ℜq2 + 4ℜp1 ℜq1)𝔍q2 - 4𝔍p2 𝔍q1
+
2                 2           2
(ℜq2  - 2ℜp1 ℜq2 + ℜp1 )𝔍q1 - 4ℜq1 𝔍p2
/
𝔍q1
+
ℜq2 + ℜp1
/
2
]
Type: List(Expression(Complex(Integer)))
axiom
#C0
(14)  2
Type: PositiveInteger?
axiom
imag(C0.1)
(15)  0
Type: Expression(Integer)
axiom
imag(C0.2)
(16)  0
Type: Expression(Integer)

axiom
ρ0:=map(x+->eval(eval(x,s1),s2),ρ)
+            - %i 𝔍q2 + ℜp1               %i 𝔍q1 + ℜq1+
|                                                     |
(17)  |(- ℜq2 + ℜp1)𝔍q2 + %i 𝔍p2 𝔍q1 - ℜq1 𝔍p2              |
|---------------------------------------  %i 𝔍q2 + ℜq2|
+                  𝔍q1                                +
Type: Matrix(Fraction(Polynomial(Complex(Integer))))
axiom
E:=eigenvalues(ρ0)
(18)
[
%D
|
2                                   2
𝔍q1 𝔍q2  + (ℜq1 ℜq2 - ℜp1 ℜq1)𝔍q2 + 𝔍p2 𝔍q1
+
2          2
((ℜp1 - %D)ℜq2 - %D ℜp1 + %D )𝔍q1 + ℜq1 𝔍p2
]
Type: List(Union(Fraction(Polynomial(Complex(Integer))),SuchThat?(Symbol,Polynomial(Complex(Integer)))))
axiom
E0:=eigenvector(E.1,ρ0)
+      %i 𝔍q1 𝔍q2 + (ℜq2 - %D)𝔍q1     +
|-------------------------------------|
(19)  [|(ℜq2 - ℜp1)𝔍q2 - %i 𝔍p2 𝔍q1 + ℜq1 𝔍p2|]
|                                     |
+                  1                  +
Type: List(Matrix(Fraction(Polynomial(Complex(Integer)))))
axiom
E1:=map(x+->eval(x,%D=C0.1),E0.1)
(20)
[
[
-
𝔍q1
*
ROOT
2                                        2
- 4𝔍q1 𝔍q2  + (- 4ℜq1 ℜq2 + 4ℜp1 ℜq1)𝔍q2 - 4𝔍p2 𝔍q1
+
2                 2           2
(ℜq2  - 2ℜp1 ℜq2 + ℜp1 )𝔍q1 - 4ℜq1 𝔍p2
/
𝔍q1
+
2%i 𝔍q1 𝔍q2 + (ℜq2 - ℜp1)𝔍q1
/
(2ℜq2 - 2ℜp1)𝔍q2 - 2%i 𝔍p2 𝔍q1 + 2ℜq1 𝔍p2
]
,
[1]]
Type: Matrix(Expression(Complex(Integer)))
axiom
E2:=map(x+->eval(x,%D=C0.2),E0.1)
(21)
[
[
𝔍q1
*
ROOT
2                                        2
- 4𝔍q1 𝔍q2  + (- 4ℜq1 ℜq2 + 4ℜp1 ℜq1)𝔍q2 - 4𝔍p2 𝔍q1
+
2                 2           2
(ℜq2  - 2ℜp1 ℜq2 + ℜp1 )𝔍q1 - 4ℜq1 𝔍p2
/
𝔍q1
+
2%i 𝔍q1 𝔍q2 + (ℜq2 - ℜp1)𝔍q1
/
(2ℜq2 - 2ℜp1)𝔍q2 - 2%i 𝔍p2 𝔍q1 + 2ℜq1 𝔍p2
]
,
[1]]
Type: Matrix(Expression(Complex(Integer)))
axiom
test(ρ0*E1=C0(1)*E1)
(22)  true
Type: Boolean
axiom
test(ρ0*E2=C0(2)*E2)
(23)  true
Type: Boolean

Given an operator , one must find the tensor for unknown manifold of hermitian isomorphisms .

axiom
h:Matrix ℂ:=matrix [[ℜa,complex(ℜb,𝔍b)],[complex(ℜb,-𝔍b),ℜe]]
+    ℜa      ℜb + 𝔍b %i+
(24)  |                      |
+ℜb - 𝔍b %i      ℜe    +
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))))
(25)  true
Type: Boolean
axiom
H:=htranspose(ρ)*h-h*ρ
(26)
[
[(- 2ℜb 𝔍p2 - 2ℜa 𝔍p1 - 2ℜp2 𝔍b)%i,
𝔍b 𝔍q2 + 𝔍b 𝔍p1 - ℜb ℜq2 - ℜa ℜq1 + ℜe ℜp2 + ℜb ℜp1
+
(- ℜb 𝔍q2 - ℜa 𝔍q1 - ℜe 𝔍p2 - ℜb 𝔍p1 + (- ℜq2 + ℜp1)𝔍b)%i
]
,
[
- 𝔍b 𝔍q2 - 𝔍b 𝔍p1 + ℜb ℜq2 + ℜa ℜq1 - ℜe ℜp2 - ℜb ℜp1
+
(- ℜb 𝔍q2 - ℜa 𝔍q1 - ℜe 𝔍p2 - ℜb 𝔍p1 + (- ℜq2 + ℜp1)𝔍b)%i
,
(- 2ℜe 𝔍q2 - 2ℜb 𝔍q1 + 2ℜq1 𝔍b)%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
J:=jacobian(concat( map(x+->[real x, imag x], concat(H::List List ?)) ),
[ℜa,ℜb,𝔍b,ℜe]::List Symbol)
+  0          0            0         0   +
|                                        |
|- 2𝔍p1    - 2𝔍p2       - 2ℜp2       0   |
|                                        |
|- ℜq1   - ℜq2 + ℜp1   𝔍q2 + 𝔍p1    ℜp2  |
|                                        |
|- 𝔍q1   - 𝔍q2 - 𝔍p1  - ℜq2 + ℜp1  - 𝔍p2 |
(27)  |                                        |
| ℜq1     ℜq2 - ℜp1   - 𝔍q2 - 𝔍p1  - ℜp2 |
|                                        |
|- 𝔍q1   - 𝔍q2 - 𝔍p1  - ℜq2 + ℜp1  - 𝔍p2 |
|                                        |
|  0          0            0         0   |
|                                        |
+  0       - 2𝔍q1        2ℜq1      - 2𝔍q2+
Type: Matrix(Fraction(Polynomial(Integer)))

The null space (kernel) of the Jacobian

axiom
N:=nullSpace(map(x+->eval(eval(x,s1),s2),J))
- ℜq2 + ℜp1 ℜq1         𝔍p2   𝔍q2
(28)  [[-----------,---,1,0],[- ---,- ---,0,1]]
𝔍q1     𝔍q1         𝔍q1   𝔍q1
Type: List(Vector(Fraction(Polynomial(Integer))))

gives the general solution to the problem.

axiom
s4:=map((x,y)+->x=y,[ℜa,ℜb,𝔍b,ℜe],𝔍b*N.1+ℜe*N.2)
- ℜe 𝔍p2 + (- ℜq2 + ℜp1)𝔍b     - ℜe 𝔍q2 + ℜq1 𝔍b
(29)  [ℜa= --------------------------,ℜb= -----------------,𝔍b= 𝔍b,ℜe= ℜe]
𝔍q1                       𝔍q1
Type: List(Equation(Fraction(Polynomial(Integer))))
axiom
map(x+->eval(eval(eval(x,s1),s2),s4),H)
+0  0+
(30)  |    |
+0  0+
Type: Matrix(Fraction(Polynomial(Complex(Integer))))

Three-Dimensions

axiom
p1:ℂ:=complex(ℜp1,𝔍p1)
(31)  ℜp1 + 𝔍p1 %i
Type: Complex(Fraction(Polynomial(Integer)))
axiom
q1:ℂ:=complex(ℜq1,𝔍q1)
(32)  ℜq1 + 𝔍q1 %i
Type: Complex(Fraction(Polynomial(Integer)))
axiom
r1:ℂ:=complex(ℜr1,𝔍r1)
(33)  ℜr1 + 𝔍r1 %i
Type: Complex(Fraction(Polynomial(Integer)))
axiom
p2:ℂ:=complex(ℜp2,𝔍p2)
(34)  ℜp2 + 𝔍p2 %i
Type: Complex(Fraction(Polynomial(Integer)))
axiom
q2:ℂ:=complex(ℜq2,𝔍q2)
(35)  ℜq2 + 𝔍q2 %i
Type: Complex(Fraction(Polynomial(Integer)))
axiom
r2:ℂ:=complex(ℜr2,𝔍r2)
(36)  ℜr2 + 𝔍r2 %i
Type: Complex(Fraction(Polynomial(Integer)))
axiom
p3:ℂ:=complex(ℜp3,𝔍p3)
(37)  ℜp3 + 𝔍p3 %i
Type: Complex(Fraction(Polynomial(Integer)))
axiom
q3:ℂ:=complex(ℜq3,𝔍q3)
(38)  ℜq3 + 𝔍q3 %i
Type: Complex(Fraction(Polynomial(Integer)))
axiom
r3:ℂ:=complex(ℜr3,𝔍r3)
(39)  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+
|                                        |
(40)  |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,ℜp3)
(41)
[
ℜp3 =
(𝔍p1 𝔍q2 - 𝔍p2 𝔍q1 - ℜp1 ℜq2 + ℜp2 ℜq1)𝔍r3
+
(- 𝔍p1 𝔍q3 + 𝔍p3 𝔍q1 + ℜp1 ℜq3)𝔍r2 + (𝔍p2 𝔍q3 - 𝔍p3 𝔍q2 - ℜp2 ℜq3)𝔍r1
+
(ℜp1 ℜr2 - ℜp2 ℜr1)𝔍q3 - ℜp1 ℜr3 𝔍q2 + ℜp2 ℜr3 𝔍q1
+
(- ℜq1 ℜr2 + ℜq2 ℜr1)𝔍p3 + (ℜq1 ℜr3 - ℜq3 ℜr1)𝔍p2
+
(- ℜq2 ℜr3 + ℜq3 ℜr2)𝔍p1
/
ℜq1 𝔍r2 - ℜq2 𝔍r1 - ℜr1 𝔍q2 + ℜr2 𝔍q1
]
Type: List(Equation(Fraction(Polynomial(Integer))))
axiom
s2:=solve(eval(imag trace(ρ),s1),𝔍p1)
(42)  [𝔍p1= - 𝔍r3 - 𝔍q2]
Type: List(Equation(Fraction(Polynomial(Integer))))
axiom
s3:=solve(eval(eval(imag trace(ρ*ρ),s1),s2),ℜp1)
(43)
[
ℜp1 =
2
- 𝔍q2 𝔍r1 𝔍r3
+
(𝔍q3 𝔍r1 + ℜq1 ℜr3)𝔍r2
+
2
(- 𝔍q2  - 𝔍p2 𝔍q1 - ℜq3 ℜr2 + ℜp2 ℜq1)𝔍r1 - ℜr1 ℜr3 𝔍q2
+
ℜr2 ℜr3 𝔍q1
*
𝔍r3
+
2
ℜq1 ℜq3 𝔍r2
+
(𝔍q2 𝔍q3 + 𝔍p3 𝔍q1 - ℜq2 ℜq3)𝔍r1 + ℜq1 ℜr2 𝔍q3
+
(- ℜq3 ℜr1 + ℜq1 ℜq2)𝔍q2 + (ℜq3 ℜr2 + ℜp2 ℜq1)𝔍q1 + ℜq1 ℜr1 𝔍p3
+
2
ℜq1 𝔍p2
*
𝔍r2
+
2
(𝔍p2 𝔍q3 - 𝔍p3 𝔍q2 - ℜp2 ℜq3)𝔍r1
+
2
(- ℜq2 ℜr2 - ℜp2 ℜr1)𝔍q3 + (ℜq2 ℜr3 - ℜq3 ℜr2 - ℜq2 )𝔍q2
+
(ℜp2 ℜr3 - ℜp2 ℜq2)𝔍q1 - ℜq1 ℜr2 𝔍p3
+
(ℜq1 ℜr3 - ℜq3 ℜr1 - ℜq1 ℜq2)𝔍p2
*
𝔍r1
+
2                     2
(- ℜr1 ℜr2 𝔍q2 + ℜr2 𝔍q1)𝔍q3 - ℜq2 ℜr1 𝔍q2
+
2                                   2
((ℜq2 ℜr2 - ℜp2 ℜr1)𝔍q1 - ℜr1 𝔍p3 - ℜq1 ℜr1 𝔍p2)𝔍q2 + ℜp2 ℜr2 𝔍q1
+
(ℜr1 ℜr2 𝔍p3 + ℜq1 ℜr2 𝔍p2)𝔍q1
/
(ℜq1 𝔍r2 - ℜr1 𝔍q2 + ℜr2 𝔍q1)𝔍r3 + (- ℜq3 𝔍r1 + ℜq1 𝔍q2)𝔍r2
+
2
(- ℜr2 𝔍q3 + (ℜr3 - ℜq2)𝔍q2)𝔍r1 - ℜr1 𝔍q2  + ℜr2 𝔍q1 𝔍q2
]
Type: List(Equation(Fraction(Polynomial(Integer))))
axiom
eval(eval(eval(imag trace(ρ*ρ*ρ),s1),s2),s3)
(44)  0
Type: Fraction(Polynomial(Integer))

axiom
C:=eval(eval(eval(characteristicPolynomial ρ,s1),s2),s3);
Type: Fraction(Polynomial(Complex(Integer)))
axiom
C0:=zerosOf(C);
Type: List(Expression(Complex(Integer)))
axiom
#C0
(47)  3
Type: PositiveInteger?
axiom
imag(C0.1)
(48)  0
Type: Expression(Integer)
axiom
imag(C0.2)
(49)  0
Type: Expression(Integer)
axiom
imag(C0.3)
(50)  0
Type: Expression(Integer)

Given an operator , 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+
|                                  |
(51)  |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)
(52)  true
Type: Boolean
axiom
H:=htranspose(ρ)*h-h*ρ
(53)
[
[(- 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
(54)
[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
--K2:=groebner(K)
J:=jacobian(select(x+->x~=0,K), [ℜa,ℜb,𝔍b,ℜc,𝔍c,ℜd,𝔍d,ℜe,ℜf]::List Symbol)
(55)
[[- 2𝔍p1,- 2𝔍p2,- 2ℜp2,- 2𝔍p3,- 2ℜp3,0,0,0,0],
[- ℜq1,- ℜq2 + ℜp1,𝔍q2 + 𝔍p1,- ℜq3,𝔍q3,ℜp3,- 𝔍p3,ℜp2,0],
[- 𝔍q1,- 𝔍q2 - 𝔍p1,- ℜq2 + ℜp1,- 𝔍q3,- ℜq3,- 𝔍p3,- ℜp3,- 𝔍p2,0],
[- ℜr1,- ℜr2,𝔍r2,- ℜr3 + ℜp1,𝔍r3 + 𝔍p1,ℜp2,𝔍p2,0,ℜp3],
[- 𝔍r1,- 𝔍r2,- ℜr2,- 𝔍r3 - 𝔍p1,- ℜr3 + ℜp1,- 𝔍p2,ℜp2,0,- 𝔍p3],
[ℜq1,ℜq2 - ℜp1,- 𝔍q2 - 𝔍p1,ℜq3,- 𝔍q3,- ℜp3,𝔍p3,- ℜp2,0],
[- 𝔍q1,- 𝔍q2 - 𝔍p1,- ℜq2 + ℜp1,- 𝔍q3,- ℜq3,- 𝔍p3,- ℜp3,- 𝔍p2,0],
[0,- 2𝔍q1,2ℜq1,0,0,- 2𝔍q3,- 2ℜq3,- 2𝔍q2,0],
[0,- ℜr1,- 𝔍r1,ℜq1,𝔍q1,- ℜr3 + ℜq2,𝔍r3 + 𝔍q2,- ℜr2,ℜq3],
[0,- 𝔍r1,ℜr1,- 𝔍q1,ℜq1,- 𝔍r3 - 𝔍q2,- ℜr3 + ℜq2,- 𝔍r2,- 𝔍q3],
[ℜr1,ℜr2,- 𝔍r2,ℜr3 - ℜp1,- 𝔍r3 - 𝔍p1,- ℜp2,- 𝔍p2,0,- ℜp3],
[- 𝔍r1,- 𝔍r2,- ℜr2,- 𝔍r3 - 𝔍p1,- ℜr3 + ℜp1,- 𝔍p2,ℜp2,0,- 𝔍p3],
[0,ℜr1,𝔍r1,- ℜq1,- 𝔍q1,ℜr3 - ℜq2,- 𝔍r3 - 𝔍q2,ℜr2,- ℜq3],
[0,- 𝔍r1,ℜr1,- 𝔍q1,ℜq1,- 𝔍r3 - 𝔍q2,- ℜr3 + ℜq2,- 𝔍r2,- 𝔍q3],
[0,0,0,- 2𝔍r1,2ℜr1,- 2𝔍r2,2ℜr2,0,- 2𝔍r3]]
Type: Matrix(Polynomial(Integer))

The null space (kernel) of the Jacobian

axiom
J2:=map(x+->eval(eval(eval(x,s1),s2),s3),J);
Type: Matrix(Fraction(Polynomial(Integer)))
axiom
nrows(J2),ncols(J2)
(57)  [15,9]
Type: Tuple(PositiveInteger?)
axiom
binomial(nrows(J2),ncols(J2))
(58)  5005
Type: PositiveInteger?
axiom
[determinant(matrix map(x+->row(J2,x+1)::List ?,subSet(nrows(J2),ncols(J2),random(binomial(nrows(J2),ncols(J2)))) )) for i in 0..1]
(59)  [0,0]
Type: List(Fraction(Polynomial(Integer)))

Eigenvectors and Diagonalization --Bill Page, Sun, 03 Jul 2011 17:35:35 -0700 reply
SandBoxHermitianIsomorphisms4

No unicode symobls --page, Mon, 04 Jul 2011 16:12:29 -0700 reply
SandBoxHermitianIsomorphism3x

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