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SandBoxHermitianIsomorphisms

Edit detail for SandBoxHermitianIsomorphisms revision 2 of 7

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Editor: Bill Page Time: 2011/06/25 19:00:53 GMT-7
Note: new

changed:
-p:=complex(Rp,Ip)
-q:=complex(Rq,Iq)
-r:=complex(Rr,Ir)
p:Complex Fraction Polynomial Integer:=complex(Rp,Ip)
q:Complex Fraction Polynomial Integer:=complex(Rq,Iq)
r:Complex Fraction Polynomial Integer:=complex(Rr,Ir)
t:Complex Fraction Polynomial Integer:=complex(Rt,0)

changed:
-h:=matrix [[a,complex(b,c)],[complex(b,-c),e]]
-matrix map(conjugate,transpose(h)::List List ?)
h:Matrix Complex Polynomial Integer:=matrix [[a,complex(b,c)],[complex(b,-c),e]]
htranspose(h)==map(x+->conjugate(x),transpose h)
test(h = htranspose h)
\end{axiom}
\begin{axiom}
H:=htranspose(ρ)*h-h*ρ
Hlist:=concat(H::List List ?)
Hreal:=map(retract,concat(map(x+->real x, Hlist),map(x+->imag x, Hlist)))
)expose MCALCFN
jacobian(Hreal,[a,b,c,e])
\end{axiom}
\begin{axiom}

axiom

p:Complex Fraction Polynomial Integer:=complex(Rp,Ip)

$\label{eq1}Rp +{Ip \ i}$

(1)

Type: Complex(Fraction(Polynomial(Integer)))

axiom

q:Complex Fraction Polynomial Integer:=complex(Rq,Iq)

$\label{eq2}Rq +{Iq \ i}$

(2)

Type: Complex(Fraction(Polynomial(Integer)))

axiom

r:Complex Fraction Polynomial Integer:=complex(Rr,Ir)

$\label{eq3}Rr +{Ir \ i}$

(3)

Type: Complex(Fraction(Polynomial(Integer)))

axiom

t:Complex Fraction Polynomial Integer:=complex(Rt,0)

$\label{eq4}Rt$

(4)

Type: Complex(Fraction(Polynomial(Integer)))

axiom

ρ:=matrix [[t/2+p,q],[r,t/2-p]]

$\label{eq5}\left[ \begin{array}{cc} {{{Rt +{2 \ Rp}}\over 2}+{Ip \ i}}&{Rq +{Iq \ i}} \ {Rr +{Ir \ i}}&{{{Rt -{2 \ Rp}}\over 2}-{Ip \ i}}$

(5)

Type: Matrix(Complex(Fraction(Polynomial(Integer))))

axiom

trace ρ

$\label{eq6}Rt$

(6)

Type: Complex(Fraction(Polynomial(Integer)))

axiom

d:=determinant ρ

$\label{eq7}\begin{array}{@{}l} \displaystyle {{{Rt^2}-{4 \ Rq \ Rr}-{4 \ {Rp^2}}+{4 \ Iq \ Ir}+{4 \ {Ip^2}}}\over 4}+ \ \ \displaystyle {{\left(-{Iq \ Rr}-{Ir \ Rq}-{2 \ Ip \ Rp}\right)}\ i}$

(7)

Type: Complex(Fraction(Polynomial(Integer)))

axiom

test(p^2+r*q=(1/4)*t^2-d)

$\label{eq8} \mbox{\rm true}$

(8)

Type: Boolean

axiom

h:Matrix Complex Polynomial Integer:=matrix [[a,complex(b,c)],[complex(b,-c),e]]

$\label{eq9}\left[ \begin{array}{cc} a &{b +{c \ i}} \ {b -{c \ i}}& e$

(9)

Type: Matrix(Complex(Polynomial(Integer)))

axiom

htranspose(h)==map(x+->conjugate(x),transpose h)

Type: Void

axiom

test(h = htranspose h)

axiom

Compiling function htranspose with type Matrix(Complex(Polynomial(
      Integer))) -> Matrix(Complex(Polynomial(Integer)))

$\label{eq10} \mbox{\rm true}$

(10)

Type: Boolean

axiom

H:=htranspose(ρ)*h-h*ρ

axiom

Compiling function htranspose with type Matrix(Complex(Fraction(
      Polynomial(Integer)))) -> Matrix(Complex(Fraction(Polynomial(
      Integer))))

$\label{eq11}\left[ \begin{array}{cc} {{\left(-{2 \ Rr \ c}-{2 \ Ir \ b}-{2 \ Ip \ a}\right)}\ i}&{{Rr \ e}+{2 \ Rp \ b}-{Rq \ a}+{{\left(-{Ir \ e}+{2 \ Rp \ c}-{Iq \ a}\right)}\ i}} \ {-{Rr \ e}-{2 \ Rp \ b}+{Rq \ a}+{{\left(-{Ir \ e}+{2 \ Rp \ c}-{Iq \ a}\right)}\ i}}&{{\left({2 \ Ip \ e}+{2 \ Rq \ c}-{2 \ Iq \ b}\right)}\ i}$

(11)

Type: Matrix(Complex(Fraction(Polynomial(Integer))))

axiom

Hlist:=concat(H::List List ?)

$\label{eq12}\begin{array}{@{}l} \displaystyle \left[{{\left(-{2 \ Rr \ c}-{2 \ Ir \ b}-{2 \ Ip \ a}\right)}\ i}, \: \right. \ \ \displaystyle \left.{{Rr \ e}+{2 \ Rp \ b}-{Rq \ a}+{{\left(-{Ir \ e}+{2 \ Rp \ c}-{Iq \ a}\right)}\ i}}, \: \right. \ \ \displaystyle \left.{-{Rr \ e}-{2 \ Rp \ b}+{Rq \ a}+{{\left(-{Ir \ e}+{2 \ Rp \ c}-{Iq \ a}\right)}\ i}}, \: \right. \ \ \displaystyle \left.{{\left({2 \ Ip \ e}+{2 \ Rq \ c}-{2 \ Iq \ b}\right)}\ i}\right]$

(12)

Type: List(Complex(Fraction(Polynomial(Integer))))

axiom

Hreal:=map(retract,concat(map(x+->real x, Hlist),map(x+->imag x, Hlist)))

   >> Error detected within library code:
   cannot retract nonconstant polynomial

axiom

A:=matrix [[Ip,Ir, Rr, 0], [Rq,-2*Rp,0,-Rr],[Iq,0,-2*Rp,Ir],[0,Iq,-Rq,-Ip]]

$\label{eq13}\left[ \begin{array}{cccc} Ip & Ir & Rr & 0 \ Rq & -{2 \ Rp}& 0 & - Rr \ Iq & 0 & -{2 \ Rp}& Ir \ 0 & Iq & - Rq & - Ip$

(13)

Type: Matrix(Polynomial(Integer))

axiom

s1:=solve(determinant(A),Rr)

$\label{eq14}\left[{Rr ={{-{Ir \ Rq}-{2 \ Ip \ Rp}}\over Iq}}\right]$

(14)

Type: List(Equation(Fraction(Polynomial(Integer))))

axiom

B:=map(x+->eval(x,s1),A)

$\label{eq15}\left[ \begin{array}{cccc} Ip & Ir &{{-{Ir \ Rq}-{2 \ Ip \ Rp}}\over Iq}& 0 \ Rq & -{2 \ Rp}& 0 &{{{Ir \ Rq}+{2 \ Ip \ Rp}}\over Iq} \ Iq & 0 & -{2 \ Rp}& Ir \ 0 & Iq & - Rq & - Ip$

(15)

Type: Matrix(Fraction(Polynomial(Integer)))

axiom

N:=nullSpace(B)

$\label{eq16}\left[{\left[{{2 \ Rp}\over Iq}, \:{Rq \over Iq}, \: 1, \: 0 \right]}, \:{\left[ -{Ir \over Iq}, \:{Ip \over Iq}, \: 0, \: 1 \right]}\right]$

(16)

Type: List(Vector(Fraction(Polynomial(Integer))))

axiom

map((x,y)+->x=y,[a,b,c,e],c*N.1+e*N.2)

$\label{eq17}\left[{a ={{-{Ir \ e}+{2 \ Rp \ c}}\over Iq}}, \:{b ={{{Ip \ e}+{Rq \ c}}\over Iq}}, \:{c = c}, \:{e = e}\right]$

(17)

Type: List(Equation(Fraction(Polynomial(Integer))))

axiom

B*(c*N(1)+e*N(2))

$\label{eq18}\left[ 0, \: 0, \: 0, \: 0 \right]$

(18)

Type: Vector(Fraction(Polynomial(Integer)))

axiom

eq27:=map((x,y)+->x=y,[a,b,c,e],c*N.1+e*N.2)

$\label{eq19}\left[{a ={{-{Ir \ e}+{2 \ Rp \ c}}\over Iq}}, \:{b ={{{Ip \ e}+{Rq \ c}}\over Iq}}, \:{c = c}, \:{e = e}\right]$

(19)

Type: List(Equation(Fraction(Polynomial(Integer))))

axiom

eq28:=map(x+->eval(x,eq27),matrix [[a,b+%I*c],[b-%I*c,e]])

$\label{eq20}\left[ \begin{array}{cc} {{-{Ir \ e}+{2 \ Rp \ c}}\over Iq}&{{{Ip \ e}+{{\left(Rq +{\%I \ Iq}\right)}\ c}}\over Iq} \ {{{Ip \ e}+{{\left(Rq -{\%I \ Iq}\right)}\ c}}\over Iq}& e$

(20)

Type: Matrix(Fraction(Polynomial(Integer)))