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last edited 13 years ago by page |
Edit detail for SandBoxHermitianIsomorphisms3 revision 5 of 11
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Editor: Bill Page
Time: 2011/06/29 16:22:25 GMT-7 |
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| Note: 3d | ||
changed: -N:=nullSpace(map(x+->eval(eval(x,s1),s2),J)) N:=nullSpace(map(x+->eval(eval(eval(x,s1),s2),s3),J))
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 $$ \rho^\dagger \circ h = h \circ \rho $$\begin{axiom} )set output tex off )set output algebra on \end{axiom}
\begin{axiom} ℂ:=Complex Fraction Polynomial Integer -- dagger htranspose(h)==map(x+->conjugate(x),transpose h) )expose MCALCFN \end{axiom}
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
\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-hρ \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) map(x+->eval(eval(eval(x,s1),s2),s4),H) \end{axiom}
Three-Dimensions \begin{axiom} p1:ℂ:=complex(ℜp1,𝔍p1) q1:ℂ:=complex(ℜq1,𝔍q1) r1:ℂ:=complex(ℜr1,𝔍r1) p2:ℂ:=complex(ℜp2,𝔍p2) q2:ℂ:=complex(ℜq2,𝔍q2) r2:ℂ:=complex(ℜr2,𝔍r2) p3:ℂ:=complex(ℜp3,𝔍p3) q3:ℂ:=complex(ℜq3,𝔍q3) r3:ℂ:=complex(ℜr3,𝔍r3) ρ:Matrix ℂ := matrix [[p1,q1,r1],[p2,q2,r2],[p3,q3,r3]] \end{axiom}
\begin{axiom} s1:=solve(imag determinant ρ,ℜp3) s2:=solve(eval(imag trace(ρ),s1),𝔍p1) s3:=solve(eval(eval(imag trace(ρ*ρ),s1),s2),ℜp1) eval(eval(eval(imag trace(ρ*ρ*ρ),s1),s2),s3) --s4:=radicalSolve(eval(eval(eval(imag trace(ρ*ρ*ρ),s1),s2),s3),𝔍q3) --#s4 --s4.1+s4.2 \end{axiom}
\begin{axiom} C:=eval(eval(eval(characteristicPolynomial ρ,s1),s2),s3); C0:=zerosOf(C); #C0 imag(C0.1) imag(C0.2) imag(C0.3) \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(ℜc,𝔍c)], _ [complex(ℜb,-𝔍b),ℜe, complex(ℜd,𝔍d)], _ [complex(ℜc,-𝔍c),complex(ℜd,-𝔍d),ℜf ]] test(h = htranspose h) H:=htranspose(ρ)h-hρ \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} K:=concat( map(x+->[real x, imag x], concat(H::List List ?)))::List Polynomial Integer --K2:=groebner(K) J:=jacobian(K, [ℜa,ℜb,𝔍b,ℜc,𝔍c,ℜd,𝔍d,ℜe,ℜf]::List Symbol) \end{axiom} The null space (kernel) of the Jacobian \begin{axiom} N:=nullSpace(map(x+->eval(eval(eval(x,s1),s2),s3),J)) \end{axiom}
Some or all expressions may not have rendered properly, because Axiom returned the following error:Error: export AXIOM=/usr/local/lib/fricas/target/x86_64-unknown-linux; export ALDORROOT=/usr/local/aldor/linux/1.1.0; export PATH=$ALDORROOT/bin:$PATH; export HOME=/var/zope2/var/LatexWiki; ulimit -t 600; export LD_LIBRARY_PATH=/usr/local/lib/fricas/target/x86_64-unknown-linux/lib; LANG=en_US.UTF-8 $AXIOM/bin/AXIOMsys < /var/zope2/var/LatexWiki/4031163534651822961-25px.axm KilledChecking for foreign routines AXIOM="/usr/local/lib/fricas/target/x86_64-unknown-linux" spad-lib="/usr/local/lib/fricas/target/x86_64-unknown-linux/lib/libspad.so" foreign routines found openServer result -2 FriCAS (AXIOM fork) Computer Algebra System Version: FriCAS 2010-12-08 Timestamp: Tuesday April 5, 2011 at 13:07:45 ----------------------------------------------------------------------------- Issue )copyright to view copyright notices. Issue )summary for a summary of useful system commands. Issue )quit to leave FriCAS and return to shell. -----------------------------------------------------------------------------
(1) -> (1) -> (1) -> (1) -> (1) -> )set output tex off
)set output algebra on
(1) -> ℂ:=Complex Fraction Polynomial Integer
(1) Complex(Fraction(Polynomial(Integer))) Type: Type -- dagger htranspose(h)==map(x+->conjugate(x),transpose h)
Type: Void )expose MCALCFN
MultiVariableCalculusFunctions is now explicitly exposed in frame initial (3) -> p1:ℂ:=complex(ℜp1,𝔍p1)
(3) ℜp1 + 𝔍p1 %i Type: Complex(Fraction(Polynomial(Integer))) q1:ℂ:=complex(ℜq1,𝔍q1)
(4) ℜq1 + 𝔍q1 %i Type: Complex(Fraction(Polynomial(Integer))) p2:ℂ:=complex(ℜp2,𝔍p2)
(5) ℜp2 + 𝔍p2 %i Type: Complex(Fraction(Polynomial(Integer))) q2:ℂ:=complex(ℜq2,𝔍q2)
(6) ℜq2 + 𝔍q2 %i Type: Complex(Fraction(Polynomial(Integer))) 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)))) (8) -> s1:=solve(imag determinant ρ,ℜp2)
ℜp1 𝔍q2 - ℜq1 𝔍p2 + ℜq2 𝔍p1 (8) [ℜp2= ---------------------------] 𝔍q1 Type: List(Equation(Fraction(Polynomial(Integer)))) s2:=solve(eval(imag trace ρ,s1),𝔍p1)
(9) [𝔍p1= - 𝔍q2] Type: List(Equation(Fraction(Polynomial(Integer)))) s3:=solve(eval(eval(imag trace(ρ*ρ),s1), s2),ℜp1)
(10) [0= 0] Type: List(Equation(Fraction(Polynomial(Integer)))) eval(eval(imag trace (ρ*ρ),s1),s2)
(11) 0 Type: Fraction(Polynomial(Integer)) (12) -> 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))) 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))) #C0
(14) 2 Type: PositiveInteger imag(C0.1)
(15) 0 Type: Expression(Integer) imag(C0.2)
(16) 0 Type: Expression(Integer) (17) -> h:Matrix ℂ:=matrix [[ℜa,complex(ℜb,𝔍b)],[complex(ℜb,-𝔍b),ℜe]]
+ ℜa ℜb + 𝔍b %i+ (17) | | +ℜb - 𝔍b %i ℜe + Type: Matrix(Complex(Fraction(Polynomial(Integer)))) test(h = htranspose h)
Compiling function htranspose with type Matrix(Complex(Fraction( Polynomial(Integer)))) -> Matrix(Complex(Fraction(Polynomial( Integer))))
(18) true Type: Boolean H:=htranspose(ρ)h-hρ
(19) [ [(- 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)))) (20) -> 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 | (20) | | | 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))) (21) -> N:=nullSpace(map(x+->eval(eval(x,s1),s2),J))
- ℜq2 + ℜp1 ℜq1 𝔍p2 𝔍q2 (21) [[-----------,---,1,0],[- ---,- ---,0,1]] 𝔍q1 𝔍q1 𝔍q1 𝔍q1 Type: List(Vector(Fraction(Polynomial(Integer)))) (22) -> 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 (22) [ℜa= --------------------------,ℜb= -----------------,𝔍b= 𝔍b,ℜe= ℜe] 𝔍q1 𝔍q1 Type: List(Equation(Fraction(Polynomial(Integer)))) map(x+->eval(eval(eval(x,s1),s2),s4),H)
+0 0+ (23) | | +0 0+ Type: Matrix(Fraction(Polynomial(Complex(Integer)))) (24) -> p1:ℂ:=complex(ℜp1,𝔍p1)
(24) ℜp1 + 𝔍p1 %i Type: Complex(Fraction(Polynomial(Integer))) q1:ℂ:=complex(ℜq1,𝔍q1)
(25) ℜq1 + 𝔍q1 %i Type: Complex(Fraction(Polynomial(Integer))) r1:ℂ:=complex(ℜr1,𝔍r1)
(26) ℜr1 + 𝔍r1 %i Type: Complex(Fraction(Polynomial(Integer))) p2:ℂ:=complex(ℜp2,𝔍p2)
(27) ℜp2 + 𝔍p2 %i Type: Complex(Fraction(Polynomial(Integer))) q2:ℂ:=complex(ℜq2,𝔍q2)
(28) ℜq2 + 𝔍q2 %i Type: Complex(Fraction(Polynomial(Integer))) r2:ℂ:=complex(ℜr2,𝔍r2)
(29) ℜr2 + 𝔍r2 %i Type: Complex(Fraction(Polynomial(Integer))) p3:ℂ:=complex(ℜp3,𝔍p3)
(30) ℜp3 + 𝔍p3 %i Type: Complex(Fraction(Polynomial(Integer))) q3:ℂ:=complex(ℜq3,𝔍q3)
(31) ℜq3 + 𝔍q3 %i Type: Complex(Fraction(Polynomial(Integer))) r3:ℂ:=complex(ℜr3,𝔍r3)
(32) ℜr3 + 𝔍r3 %i Type: Complex(Fraction(Polynomial(Integer))) ρ:Matrix ℂ := matrix [[p1,q1,r1],[p2,q2,r2],[p3,q3,r3]]
+ℜp1 + 𝔍p1 %i ℜq1 + 𝔍q1 %i ℜr1 + 𝔍r1 %i+ | | (33) |ℜp2 + 𝔍p2 %i ℜq2 + 𝔍q2 %i ℜr2 + 𝔍r2 %i| | | +ℜp3 + 𝔍p3 %i ℜq3 + 𝔍q3 %i ℜr3 + 𝔍r3 %i+ Type: Matrix(Complex(Fraction(Polynomial(Integer)))) (34) -> s1:=solve(imag determinant ρ,ℜp3)
(34) [ 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)))) s2:=solve(eval(imag trace(ρ),s1),𝔍p1)
(35) [𝔍p1= - 𝔍r3 - 𝔍q2] Type: List(Equation(Fraction(Polynomial(Integer)))) s3:=solve(eval(eval(imag trace(ρ*ρ),s1),s2),ℜp1)
(36) [ 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)))) eval(eval(eval(imag trace(ρ*ρ*ρ),s1),s2),s3)
(37) 0 Type: Fraction(Polynomial(Integer)) (38) -> C:=eval(eval(eval(characteristicPolynomial ρ,s1),s2),s3);
Type: Fraction(Polynomial(Complex(Integer))) C0:=zerosOf(C);
Type: List(Expression(Complex(Integer))) #C0
(40) 3 Type: PositiveInteger imag(C0.1)
(41) 0 Type: Expression(Integer) imag(C0.2)
(42) 0 Type: Expression(Integer) imag(C0.3)
(43) 0 Type: Expression(Integer) (44) -> 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+ | | (44) |ℜb - 𝔍b %i ℜe ℜd + 𝔍d %i| | | +ℜc - 𝔍c %i ℜd - 𝔍d %i ℜf + Type: Matrix(Complex(Fraction(Polynomial(Integer)))) test(h = htranspose h)
(45) true Type: Boolean H:=htranspose(ρ)h-hρ
(46) [ [(- 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)))) (47) -> K:=concat( map(x+->[real x, imag x], concat(H::List List ?)))::List Polynomial Integer
(47) [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)) --K2:=groebner(K) J:=jacobian(K, [ℜa,ℜb,𝔍b,ℜc,𝔍c,ℜd,𝔍d,ℜe,ℜf]::List Symbol)
(48) [[0,0,0,0,0,0,0,0,0], [- 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], [0,0,0,0,0,0,0,0,0], [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], [0,0,0,0,0,0,0,0,0], [0,0,0,- 2𝔍r1,2ℜr1,- 2𝔍r2,2ℜr2,0,- 2𝔍r3]] Type: Matrix(Polynomial(Integer)) (49) -> N:=nullSpace(map(x+->eval(eval(eval(x,s1),s2),s3),J))
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(/usr/share/texmf-texlive/tex/latex/ucs/data/uni-33.def) (/usr/share/texmf-texlive/tex/latex/bbm/ubbm.fd)
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(/usr/share/texmf-texlive/tex/latex/ucs/data/uni-3.def) [6] [7] [8] [9] (/usr/share/texmf-texlive/tex/latex/ucs/data/uni-469.def) (/usr/share/texmf-texlive/tex/latex/amsfonts/ueuf.fd) Missing $ inserted. <inserted text> $ l.148 ρ:Matrix ℂ := matrix [[p1,q1],[p2,q2]]
Missing $ inserted. <inserted text> $ l.148 ρ:Matrix ℂ := matrix [[p1,q1],[p2,q2]]
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Missing $ inserted. <inserted text> $ l.151 s1:=solve(imag determinant ρ,ℜp2)
Missing $ inserted. <inserted text> $ l.151 s1:=solve(imag determinant ρ,ℜp2)
Missing $ inserted. <inserted text> $ l.152 s2:=solve(eval(imag trace ρ,s1),𝔍p1)
Missing $ inserted. <inserted text> $ l.152 s2:=solve(eval(imag trace ρ,s1),𝔍p1)
Missing $ inserted. <inserted text> $ l.153 ...al(eval(imag trace(ρ*ρ),s1), s2),ℜp1)
Missing $ inserted. <inserted text> $ l.153 ...al(eval(imag trace(ρ*ρ),s1), s2),ℜp1)
Missing $ inserted. <inserted text> $ l.154 eval(eval(imag trace (ρ*ρ),s1),s2)
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Missing $ inserted. <inserted text> $ l.157 ...(eval(characteristicPolynomial ρ,s1),s2)
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[10] [11] [12] Missing $ inserted. <inserted text> $ l.169 H:=htranspose(ρ)h-hρ
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Missing $ inserted. <inserted text> $ l.196 ...matrix [[p1,q1,r1],[p2,q2,r2],[p3,q3,r3]]
Missing $ inserted. <inserted text> $ l.196 ...matrix [[p1,q1,r1],[p2,q2,r2],[p3,q3,r3]]
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Missing $ inserted. <inserted text> $ l.199 s1:=solve(imag determinant ρ,ℜp3)
Missing $ inserted. <inserted text> $ l.199 s1:=solve(imag determinant ρ,ℜp3)
Missing $ inserted. <inserted text> $ l.200 s2:=solve(eval(imag trace(ρ),s1),𝔍p1)
Missing $ inserted. <inserted text> $ l.200 s2:=solve(eval(imag trace(ρ),s1),𝔍p1)
Missing $ inserted. <inserted text> $ l.201 ...val(eval(imag trace(ρ*ρ),s1),s2),ℜp1)
Missing $ inserted. <inserted text> $ l.201 ...val(eval(imag trace(ρ*ρ),s1),s2),ℜp1)
Missing $ inserted. <inserted text> $ l.202 ...val(eval(imag trace(ρ*ρ*ρ),s1),s2),s3)
Missing $ inserted. <inserted text> $ l.202 ...val(eval(imag trace(ρ*ρ*ρ),s1),s2),s3)
Missing $ inserted. <inserted text> $ l.203 ...(imag trace(ρ*ρ*ρ),s1),s2),s3),𝔍q3)
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Missing $ inserted. <inserted text> $ l.208 ...(characteristicPolynomial ρ,s1),s2),s3);
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[17] [18] [19] Missing $ inserted. <inserted text> $ l.223 H:=htranspose(ρ)h-hρ
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[24] (./7645513574781919064-16.0px.aux) ) (see the transcript file for additional information) Output written on 7645513574781919064-16.0px.dvi (24 pages, 8312 bytes). Transcript written on 7645513574781919064-16.0px.log.