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 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} \begin{axiom} ρ0:=map(x+->eval(eval(x,s1),s2),ρ) E:=eigenvalues(ρ0) E0:=eigenvector(E.1,ρ0) E1:=map(x+->eval(x,%D=C0.1),E0.1) E2:=map(x+->eval(x,%D=C0.2),E0.1) test(ρ0E1=C0(1)E1) test(ρ0E2=C0(2)E2) \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(select(x+->x~=0,K), [ℜa,ℜb,𝔍b,ℜc,𝔍c,ℜd,𝔍d,ℜe,ℜf]::List Symbol) \end{axiom} The null space (kernel) of the Jacobian \begin{axiom} J2:=map(x+->eval(eval(eval(x,s1),s2),s3),J); nrows(J2),ncols(J2) binomial(nrows(J2),ncols(J2)) [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] --N:=nullSpace J2 \end{axiom} Eigenvectors and Diagonalization --Bill Page, Sun, 03 Jul 2011 17:35:35 -0700 replySandBoxHermitianIsomorphisms4 No unicode symobls --page, Mon, 04 Jul 2011 16:12:29 -0700 reply[SandBoxHermitianIsomorphism3x] Some or all expressions may not have rendered properly, because Axiom returned the following error:Error: export FRICAS=/usr/local/lib/fricas/target/x86_64-linux-gnu; 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-linux-gnu/lib; LANG=en_US.UTF-8 $FRICAS/bin/FRICASsys < /var/lib/zope2.10/instance/axiom-wiki/var/LatexWiki/6375190464244529648-25px.axm Killed Checking for foreign routines FRICAS="/usr/local/lib/fricas/target/x86_64-linux-gnu" spad-lib="/usr/local/lib/fricas/target/x86_64-linux-gnu/lib/libspad.so" foreign routines found openServer result -2 FriCAS Computer Algebra System Version: FriCAS 1.3.7 Timestamp: Wed Jun 30 16:44:06 UTC 2021 ----------------------------------------------------------------------------- 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 - %B)ℜq2 - %B ℜp1 + %B )𝔍q1 + ℜq1 𝔍p2 / 𝔍q1 Type: Fraction(Polynomial(Complex(Integer))) C0:=zerosOf(C) (13) [ ROOT 2 2 - 4 𝔍q1 𝔍q2 + (- 4 ℜq1 ℜq2 + 4 ℜp1 ℜq1)𝔍q2 - 4 𝔍p2 𝔍q1 + 2 2 2 (ℜq2 - 2 ℜp1 ℜq2 + ℜp1 )𝔍q1 - 4 ℜq1 𝔍p2 / 𝔍q1 + ℜq2 + ℜp1 / 2 , ROOT 2 2 - 4 𝔍q1 𝔍q2 + (- 4 ℜq1 ℜq2 + 4 ℜp1 ℜq1)𝔍q2 - 4 𝔍p2 𝔍q1 + 2 2 2 (ℜq2 - 2 ℜp1 ℜq2 + ℜp1 )𝔍q1 - 4 ℜq1 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) -> 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)))) 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))))) 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))))) E1:=map(x+->eval(x,%D=C0.1),E0.1) (20) [ [ - 𝔍q1 * ROOT 2 2 - 4 𝔍q1 𝔍q2 + (- 4 ℜq1 ℜq2 + 4 ℜp1 ℜq1)𝔍q2 - 4 𝔍p2 𝔍q1 + 2 2 2 (ℜq2 - 2 ℜp1 ℜq2 + ℜp1 )𝔍q1 - 4 ℜq1 𝔍p2 / 𝔍q1 + 2 %i 𝔍q1 𝔍q2 + (ℜq2 - ℜp1)𝔍q1 / (2 ℜq2 - 2 ℜp1)𝔍q2 - 2 %i 𝔍p2 𝔍q1 + 2 ℜq1 𝔍p2 ] , [1]] Type: Matrix(Expression(Complex(Integer))) E2:=map(x+->eval(x,%D=C0.2),E0.1) (21) [ [ 𝔍q1 * ROOT 2 2 - 4 𝔍q1 𝔍q2 + (- 4 ℜq1 ℜq2 + 4 ℜp1 ℜq1)𝔍q2 - 4 𝔍p2 𝔍q1 + 2 2 2 (ℜq2 - 2 ℜp1 ℜq2 + ℜp1 )𝔍q1 - 4 ℜq1 𝔍p2 / 𝔍q1 + 2 %i 𝔍q1 𝔍q2 + (ℜq2 - ℜp1)𝔍q1 / (2 ℜq2 - 2 ℜp1)𝔍q2 - 2 %i 𝔍p2 𝔍q1 + 2 ℜq1 𝔍p2 ] , [1]] Type: Matrix(Expression(Complex(Integer))) test(ρ0E1=C0(1)E1) (22) true Type: Boolean test(ρ0E2=C0(2)E2) (23) true Type: Boolean (24) -> 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)))) test(h = htranspose h) Compiling function htranspose with type Matrix(Complex(Fraction( Polynomial(Integer)))) -> Matrix(Complex(Fraction(Polynomial( Integer)))) (25) true Type: Boolean H:=htranspose(ρ)h-hρ (26) [ [(- 2 ℜb 𝔍p2 - 2 ℜa 𝔍p1 - 2 ℜp2 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 , (- 2 ℜe 𝔍q2 - 2 ℜb 𝔍q1 + 2 ℜq1 𝔍b)%i] ] Type: Matrix(Complex(Fraction(Polynomial(Integer)))) (27) -> J:=jacobian(concat( map(x+->[real x, imag x], concat(H::List List ?)) ), [ℜa,ℜb,𝔍b,ℜe]::List Symbol) + 0 0 0 0 + | | |- 2 𝔍p1 - 2 𝔍p2 - 2 ℜp2 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 - 2 𝔍q1 2 ℜq1 - 2 𝔍q2+ Type: Matrix(Fraction(Polynomial(Integer))) (28) -> 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)))) (29) -> s4:=map((x,y)+->x=y,[ℜa,ℜb,𝔍b,ℜe],𝔍b*N.1+ℜe*N.2) (29) - ℜe 𝔍p2 + (- ℜq2 + ℜp1)𝔍b - ℜe 𝔍q2 + ℜq1 𝔍b [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+ (30) | | +0 0+ Type: Matrix(Fraction(Polynomial(Complex(Integer)))) (31) -> p1:ℂ:=complex(ℜp1,𝔍p1) (31) ℜp1 + 𝔍p1 %i Type: Complex(Fraction(Polynomial(Integer))) q1:ℂ:=complex(ℜq1,𝔍q1) (32) ℜq1 + 𝔍q1 %i Type: Complex(Fraction(Polynomial(Integer))) r1:ℂ:=complex(ℜr1,𝔍r1) (33) ℜr1 + 𝔍r1 %i Type: Complex(Fraction(Polynomial(Integer))) p2:ℂ:=complex(ℜp2,𝔍p2) (34) ℜp2 + 𝔍p2 %i Type: Complex(Fraction(Polynomial(Integer))) q2:ℂ:=complex(ℜq2,𝔍q2) (35) ℜq2 + 𝔍q2 %i Type: Complex(Fraction(Polynomial(Integer))) r2:ℂ:=complex(ℜr2,𝔍r2) (36) ℜr2 + 𝔍r2 %i Type: Complex(Fraction(Polynomial(Integer))) p3:ℂ:=complex(ℜp3,𝔍p3) (37) ℜp3 + 𝔍p3 %i Type: Complex(Fraction(Polynomial(Integer))) q3:ℂ:=complex(ℜq3,𝔍q3) (38) ℜq3 + 𝔍q3 %i Type: Complex(Fraction(Polynomial(Integer))) r3:ℂ:=complex(ℜr3,𝔍r3) (39) 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+ | | (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)))) (41) -> 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)))) s2:=solve(eval(imag trace(ρ),s1),𝔍p1) (42) [𝔍p1 = - 𝔍r3 - 𝔍q2] Type: List(Equation(Fraction(Polynomial(Integer)))) 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)))) eval(eval(eval(imag trace(ρ*ρ*ρ),s1),s2),s3) (44) 0 Type: Fraction(Polynomial(Integer)) (45) -> C:=eval(eval(eval(characteristicPolynomial ρ,s1),s2),s3); Type: Fraction(Polynomial(Complex(Integer))) C0:=zerosOf(C); Type: List(Expression(Complex(Integer))) #C0 (47) 3 Type: PositiveInteger imag(C0.1) (48) 0 Type: Expression(Integer) imag(C0.2) (49) 0 Type: Expression(Integer) imag(C0.3) (50) 0 Type: Expression(Integer) (51) -> 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)))) test(h = htranspose h) (52) true Type: Boolean H:=htranspose(ρ)h-hρ (53) [ [(- 2 ℜc 𝔍p3 - 2 ℜb 𝔍p2 - 2 ℜa 𝔍p1 - 2 ℜp3 𝔍c - 2 ℜp2 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 , (- 2 ℜd 𝔍q3 - 2 ℜe 𝔍q2 - 2 ℜb 𝔍q1 - 2 ℜq3 𝔍d + 2 ℜq1 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 , (- 2 ℜf 𝔍r3 - 2 ℜd 𝔍r2 - 2 ℜc 𝔍r1 + 2 ℜr2 𝔍d + 2 ℜr1 𝔍c)%i] ] Type: Matrix(Complex(Fraction(Polynomial(Integer)))) (54) -> K:=concat( map(x+->[real x, imag x], concat(H::List List ?)))::List Polynomial Integer (54) [0, - 2 ℜc 𝔍p3 - 2 ℜb 𝔍p2 - 2 ℜa 𝔍p1 - 2 ℜp3 𝔍c - 2 ℜp2 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, - 2 ℜd 𝔍q3 - 2 ℜe 𝔍q2 - 2 ℜb 𝔍q1 - 2 ℜq3 𝔍d + 2 ℜq1 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, - 2 ℜf 𝔍r3 - 2 ℜd 𝔍r2 - 2 ℜc 𝔍r1 + 2 ℜr2 𝔍d + 2 ℜr1 𝔍c] Type: List(Polynomial(Integer)) --K2:=groebner(K) J:=jacobian(select(x+->x~=0,K), [ℜa,ℜb,𝔍b,ℜc,𝔍c,ℜd,𝔍d,ℜe,ℜf]::List Symbol) (55) [[- 2 𝔍p1, - 2 𝔍p2, - 2 ℜp2, - 2 𝔍p3, - 2 ℜp3, 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, - 2 𝔍q1, 2 ℜq1, 0, 0, - 2 𝔍q3, - 2 ℜq3, - 2 𝔍q2, 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, - 2 𝔍r1, 2 ℜr1, - 2 𝔍r2, 2 ℜr2, 0, - 2 𝔍r3]] Type: Matrix(Polynomial(Integer)) (56) -> J2:=map(x+->eval(eval(eval(x,s1),s2),s3),J); Type: Matrix(Fraction(Polynomial(Integer))) nrows(J2),ncols(J2) (57) [15, 9] Type: Tuple(PositiveInteger) binomial(nrows(J2),ncols(J2)) (58) 5005 Type: PositiveInteger [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]  Some or all expressions may not have rendered properly, because Latex returned the following error:! Missing$ inserted. $l.147 ρ:Matrix ℂ := matrix [[p1,q1],[p2,q2]] Missing$ inserted. $l.147 ρ:Matrix ℂ := matrix [[p1,q1],[p2,q2]] LaTeX Warning: Characters dropped after \end{axiom}' on input line 148. Missing$ inserted. $l.150 s1:=solve(imag determinant ρ,ℜp2) Missing$ inserted. $l.150 s1:=solve(imag determinant ρ,ℜp2) Missing$ inserted. $l.151 s2:=solve(eval(imag trace ρ,s1),𝔍p1) Missing$ inserted. $l.151 s2:=solve(eval(imag trace ρ,s1),𝔍p1) Missing$ inserted. $l.152 ...al(eval(imag trace(ρ*ρ),s1), s2),ℜp1) Missing$ inserted. $l.152 ...al(eval(imag trace(ρ*ρ),s1), s2),ℜp1) Missing$ inserted. $l.153 eval(eval(imag trace (ρ*ρ),s1),s2) Missing$ inserted. $l.153 eval(eval(imag trace (ρ*ρ),s1),s2) LaTeX Warning: Characters dropped after \end{axiom}' on input line 154. Missing$ inserted. $l.156 ...(eval(characteristicPolynomial ρ,s1),s2) Missing$ inserted. $l.156 ...(eval(characteristicPolynomial ρ,s1),s2) LaTeX Warning: Characters dropped after \end{axiom}' on input line 161. Missing$ inserted. $l.163 ρ0:=map(x+->eval(eval(x,s1),s2),ρ) Missing$ inserted. $l.163 ρ0:=map(x+->eval(eval(x,s1),s2),ρ) Missing$ inserted. $l.164 E:=eigenvalues(ρ0) Missing$ inserted. $l.164 E:=eigenvalues(ρ0) Missing$ inserted. $l.165 E0:=eigenvector(E.1,ρ0) Missing$ inserted. $l.165 E0:=eigenvector(E.1,ρ0) Missing$ inserted. $l.168 test(ρ0E1=C0(1)E1) Missing$ inserted. $l.168 test(ρ0E1=C0(1)E1) Missing$ inserted. $l.169 test(ρ0E2=C0(2)E2) Missing$ inserted. $l.169 test(ρ0E2=C0(2)E2) LaTeX Warning: Characters dropped after \end{axiom}' on input line 170. [10] [11] [12] Missing$ inserted. $l.177 H:=htranspose(ρ)h-hρ Missing$ inserted. $l.177 H:=htranspose(ρ)h-hρ LaTeX Warning: Characters dropped after \end{axiom}' on input line 178. [13] [14] [15] [16] LaTeX Warning: Characters dropped after \end{axiom}' on input line 186. LaTeX Warning: Characters dropped after \end{axiom}' on input line 189. LaTeX Warning: Characters dropped after \end{axiom}' on input line 193. Missing$ inserted. $l.204 ...matrix [[p1,q1,r1],[p2,q2,r2],[p3,q3,r3]] Missing$ inserted. $l.204 ...matrix [[p1,q1,r1],[p2,q2,r2],[p3,q3,r3]] LaTeX Warning: Characters dropped after \end{axiom}' on input line 205. Missing$ inserted. $l.207 s1:=solve(imag determinant ρ,ℜp3) Missing$ inserted. $l.207 s1:=solve(imag determinant ρ,ℜp3) Missing$ inserted. $l.208 s2:=solve(eval(imag trace(ρ),s1),𝔍p1) Missing$ inserted. $l.208 s2:=solve(eval(imag trace(ρ),s1),𝔍p1) Missing$ inserted. $l.209 ...val(eval(imag trace(ρ*ρ),s1),s2),ℜp1) Missing$ inserted. $l.209 ...val(eval(imag trace(ρ*ρ),s1),s2),ℜp1) Missing$ inserted. $l.210 ...val(eval(imag trace(ρ*ρ*ρ),s1),s2),s3) Missing$ inserted. $l.210 ...val(eval(imag trace(ρ*ρ*ρ),s1),s2),s3) Missing$ inserted. $l.211 ...(imag trace(ρ*ρ*ρ),s1),s2),s3),𝔍q3) Missing$ inserted. $l.211 ...(imag trace(ρ*ρ*ρ),s1),s2),s3),𝔍q3) LaTeX Warning: Characters dropped after \end{axiom}' on input line 214. Missing$ inserted. $l.216 ...(characteristicPolynomial ρ,s1),s2),s3); Missing$ inserted. $l.216 ...(characteristicPolynomial ρ,s1),s2),s3); LaTeX Warning: Characters dropped after \end{axiom}' on input line 222. [17] [18] [19] Missing$ inserted. $l.231 H:=htranspose(ρ)h-hρ Missing$ inserted. \$ l.231 H:=htranspose(ρ)h-hρ LaTeX Warning: Characters dropped after \end{axiom}' on input line 232. [20] [21] [22] [23] LaTeX Warning: Characters dropped after \end{axiom}' on input line 241. Overfull \hbox (86.02751pt too wide) in paragraph at lines 246--246 []\T1/cmtt/m/n/12 [determinant(matrix map(x+->row(J2,x+1)::List ?`

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