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SandBoxHermitianIsomorphisms

Edit detail for SandBoxHermitianIsomorphisms revision 1 of 7

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Editor: Bill Page Time: 2011/06/25 17:36:05 GMT-7
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changed:
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\begin{axiom}
p:=complex(Rp,Ip)
q:=complex(Rq,Iq)
r:=complex(Rr,Ir)
ρ:=matrix [[t/2+p,q],[r,t/2-p]]
trace ρ
d:=determinant ρ
test(p^2+r*q=(1/4)*t^2-d)
\end{axiom}
\begin{axiom}
h:=matrix [[a,complex(b,c)],[complex(b,-c),e]]
matrix map(conjugate,transpose(h)::List List ?)
A:=matrix [[Ip,Ir, Rr, 0], [Rq,-2*Rp,0,-Rr],[Iq,0,-2*Rp,Ir],[0,Iq,-Rq,-Ip]]
s1:=solve(determinant(A),Rr)                                               
B:=map(x+->eval(x,s1),A)                                                   
N:=nullSpace(B)                                                            
map((x,y)+->x=y,[a,b,c,e],c*N.1+e*N.2)  
B*(c*N(1)+e*N(2))
eq27:=map((x,y)+->x=y,[a,b,c,e],c*N.1+e*N.2)
eq28:=map(x+->eval(x,eq27),matrix [[a,b+%I*c],[b-%I*c,e]])  
\end{axiom}

axiom

p:=complex(Rp,Ip)

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

(1)

Type: Complex(Polynomial(Integer))

axiom

q:=complex(Rq,Iq)

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

(2)

Type: Complex(Polynomial(Integer))

axiom

r:=complex(Rr,Ir)

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

(3)

Type: Complex(Polynomial(Integer))

axiom

ρ:=matrix [[t/2+p,q],[r,t/2-p]]

$\label{eq4}\left[ \begin{array}{cc} {{{1 \over 2}\ t}+ Rp +{i \ Ip}}&{Rq +{i \ Iq}} \ {Rr +{i \ Ir}}&{{{1 \over 2}\ t}- Rp -{i \ Ip}}$

(4)

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

axiom

trace ρ

$\label{eq5}t$

(5)

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

axiom

d:=determinant ρ

$\label{eq6}\begin{array}{@{}l} \displaystyle {{1 \over 4}\ {t^2}}+{{\left(- Rq -{i \ Iq}\right)}\ Rr}-{i \ Ir \ Rq}-{Rp^2}-{2 \ i \ Ip \ Rp}+ \ \ \displaystyle {Iq \ Ir}+{Ip^2}$

(6)

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

axiom

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

$\label{eq7} \mbox{\rm true}$

(7)

Type: Boolean

axiom

h:=matrix [[a,complex(b,c)],[complex(b,-c),e]]

$\label{eq8}\left[ \begin{array}{cc} a &{{i \ c}+ b} \ {-{i \ c}+ b}& e$

(8)

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

axiom

matrix map(conjugate,transpose(h)::List List ?)

   There are 67 exposed and 8 unexposed library operations named map 
      having 2 argument(s) but none was determined to be applicable. 
      Use HyperDoc Browse, or issue
                               )display op map
      to learn more about the available operations. Perhaps 
      package-calling the operation or using coercions on the arguments
      will allow you to apply the operation.

   Cannot find a definition or applicable library operation named map 
      with argument type(s) 
                             Variable(conjugate)
                  List(List(Polynomial(Complex(Integer))))

      Perhaps you should use "@" to indicate the required return type, 
      or "$" to specify which version of the function you need.
A:=matrix [[Ip,Ir, Rr, 0], [Rq,-2*Rp,0,-Rr],[Iq,0,-2*Rp,Ir],[0,Iq,-Rq,-Ip]]

$\label{eq9}\left[ \begin{array}{cccc} Ip & Ir & Rr & 0 \ Rq & -{2 \ Rp}& 0 & - Rr \ Iq & 0 & -{2 \ Rp}& Ir \ 0 & Iq & - Rq & - Ip$

(9)

Type: Matrix(Polynomial(Integer))

axiom

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

$\label{eq10}\left[{Rr ={{-{Ir \ Rq}-{2 \ Ip \ Rp}}\over Iq}}\right]$

(10)

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

axiom

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

$\label{eq11}\left[ \begin{array}{cccc} Ip & Ir &{{-{Ir \ Rq}-{2 \ Ip \ Rp}}\over Iq}& 0 \ Rq & -{2 \ Rp}& 0 &{{{Ir \ Rq}+{2 \ Ip \ Rp}}\over Iq} \ Iq & 0 & -{2 \ Rp}& Ir \ 0 & Iq & - Rq & - Ip$

(11)

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

axiom

N:=nullSpace(B)

$\label{eq12}\left[{\left[{{2 \ Rp}\over Iq}, \:{Rq \over Iq}, \: 1, \: 0 \right]}, \:{\left[ -{Ir \over Iq}, \:{Ip \over Iq}, \: 0, \: 1 \right]}\right]$

(12)

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

axiom

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

$\label{eq13}\left[{a ={{-{Ir \ e}+{2 \ Rp \ c}}\over Iq}}, \:{b ={{{Ip \ e}+{Rq \ c}}\over Iq}}, \:{c = c}, \:{e = e}\right]$

(13)

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

axiom

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

$\label{eq14}\left[ 0, \: 0, \: 0, \: 0 \right]$

(14)

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

axiom

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

$\label{eq15}\left[{a ={{-{Ir \ e}+{2 \ Rp \ c}}\over Iq}}, \:{b ={{{Ip \ e}+{Rq \ c}}\over Iq}}, \:{c = c}, \:{e = e}\right]$

(15)

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

axiom

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

$\label{eq16}\left[ \begin{array}{cc} {{-{Ir \ e}+{2 \ Rp \ c}}\over Iq}&{{{Ip \ e}+{{\left(Rq +{\%I \ Iq}\right)}\ c}}\over Iq} \ {{{Ip \ e}+{{\left(Rq -{\%I \ Iq}\right)}\ c}}\over Iq}& e$

(16)

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