A complex vector ℂ-space possesses many different hermitian isomorphisms
. In quantum mechanics a given operator
may be said to be -hermitian if
axiom
ℂ:=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)
Type: Complex(Fraction(Polynomial(Integer)))
axiom
q1:ℂ:=complex(ℜq1,𝔍q1)
Type: Complex(Fraction(Polynomial(Integer)))
axiom
p2:ℂ:=complex(ℜp2,𝔍p2)
Type: Complex(Fraction(Polynomial(Integer)))
axiom
q2:ℂ:=complex(ℜq2,𝔍q2)
Type: Complex(Fraction(Polynomial(Integer)))
axiom
ρ:Matrix ℂ := matrix [[p1,q1],[p2,q2]]
Type: Matrix(Complex(Fraction(Polynomial(Integer))))
axiom
s1:=solve(imag determinant ρ,ℜp2)
Type: List(Equation(Fraction(Polynomial(Integer))))
axiom
s2:=solve(eval(imag trace ρ, s1),𝔍q2)
Type: List(Equation(Fraction(Polynomial(Integer))))
axiom
eval(eval(imag trace (ρ*ρ),s1),s2)
Type: Fraction(Polynomial(Integer))
Given an operator , one must find the tensor
for unknown manifold of hermitian isomorphisms .
axiom
h:Matrix ℂ:=matrix [[a,complex(b,c)],[complex(b,-c),e]]
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))))
Type: Boolean
axiom
H:=htranspose(ρ)*h-h*ρ
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)
Type: Matrix(Fraction(Polynomial(Integer)))
The null space (kernel) of the Jacobian
axiom
N:=nullSpace(map(x+->eval(eval(x,s1),s2),J))
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)
Type: List(Equation(Fraction(Polynomial(Integer))))
axiom
map(x+->eval(eval(eval(x,s1),s2),s3),H)
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 , 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]
The null space (kernel) of the Jacobian
Axiom output parse error!