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last edited 12 years ago by Bill Page |
Edit detail for SandBoxHermitianIsomorphisms revision 4 of 7
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Editor: Bill Page
Time: 2011/06/26 11:49:55 GMT-7 |
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| Note: jacobian | ||
changed: -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]]) s2:=map((x,y)+->x=y,[a,b,c,e],c*N.1+e*N.2) h1:=map(x+->eval(x,s2),h) map(x+->eval(x,s1),htranspose(ρ)*h1-h1*ρ)
axiom
p:Complex Fraction Polynomial Integer:=complex(Rp,Ip)
 | (1) |
Type: Complex(Fraction(Polynomial(Integer)))
axiom
q:Complex Fraction Polynomial Integer:=complex(Rq,Iq)
 | (2) |
Type: Complex(Fraction(Polynomial(Integer)))
axiom
r:Complex Fraction Polynomial Integer:=complex(Rr,Ir)
 | (3) |
Type: Complex(Fraction(Polynomial(Integer)))
axiom
t:Complex Fraction Polynomial Integer:=complex(Rt,0)
 | (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}}](images/3675818863918649304-16.0px.png "\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
 | (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}](images/7191737606198833646-16.0px.png "\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)
 | (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](images/932627303795594587-16.0px.png "\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))) | (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}](images/9172979035813233560-16.0px.png "\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]](images/3445897294084933332-16.0px.png "\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:=removeDuplicates(select(x+->(x~=0),concat(map(x+->real x, Hlist), map(x+->imag x, Hlist))))
![\label{eq13}\begin{array}{@{}l}
\displaystyle
\left[{{Rr \ e}+{2 \ Rp \ b}-{Rq \ a}}, \:{-{Rr \ e}-{2 \ Rp \ b}+{Rq \ a}}, \: \right.
\
\
\displaystyle
\left.{-{2 \ Rr \ c}-{2 \ Ir \ b}-{2 \ Ip \ a}}, \:{-{Ir \ e}+{2 \ Rp \ c}-{Iq \ a}}, \: \right.
\
\
\displaystyle
\left.{{2 \ Ip \ e}+{2 \ Rq \ c}-{2 \ Iq \ b}}\right]](images/1322605763096927188-16.0px.png "\label{eq13}\begin{array}{@{}l}
\displaystyle
\left[{{Rr \ e}+{2 \ Rp \ b}-{Rq \ a}}, \:{-{Rr \ e}-{2 \ Rp \ b}+{Rq \ a}}, \: \right.
\
\
\displaystyle
\left.{-{2 \ Rr \ c}-{2 \ Ir \ b}-{2 \ Ip \ a}}, \:{-{Ir \ e}+{2 \ Rp \ c}-{Iq \ a}}, \: \right.
\
\
\displaystyle
\left.{{2 \ Ip \ e}+{2 \ Rq \ c}-{2 \ Iq \ b}}\right]") | (13) |
Type: List(Fraction(Polynomial(Integer)))
axiom
)expose MCALCFN
MultiVariableCalculusFunctions is now explicitly exposed in frame initial H1:=jacobian(Hreal,[a, b, c, e]::List Symbol)
![\label{eq14}\left[
\begin{array}{cccc}
- Rq &{2 \ Rp}& 0 & Rr
\
Rq & -{2 \ Rp}& 0 & - Rr
\
-{2 \ Ip}& -{2 \ Ir}& -{2 \ Rr}& 0
\
- Iq & 0 &{2 \ Rp}& - Ir
\
0 & -{2 \ Iq}&{2 \ Rq}&{2 \ Ip}](images/473588788701647089-16.0px.png "\label{eq14}\left[
\begin{array}{cccc}
- Rq &{2 \ Rp}& 0 & Rr
\
Rq & -{2 \ Rp}& 0 & - Rr
\
-{2 \ Ip}& -{2 \ Ir}& -{2 \ Rr}& 0
\
- Iq & 0 &{2 \ Rp}& - Ir
\
0 & -{2 \ Iq}&{2 \ Rq}&{2 \ Ip}") | (14) |
Type: Matrix(Fraction(Polynomial(Integer)))
axiom
s1:=solve(determinant subMatrix(H1,2, 5, 1, 4), Rr)
![\label{eq15}\left[{Rr ={{-{Ir \ Rq}-{2 \ Ip \ Rp}}\over Iq}}\right]](images/4592615677198746218-16.0px.png "\label{eq15}\left[{Rr ={{-{Ir \ Rq}-{2 \ Ip \ Rp}}\over Iq}}\right]") | (15) |
Type: List(Equation(Fraction(Polynomial(Integer))))
axiom
H2:=map(x+->eval(x,s1), H1)
![\label{eq16}\left[
\begin{array}{cccc}
- Rq &{2 \ Rp}& 0 &{{-{Ir \ Rq}-{2 \ Ip \ Rp}}\over Iq}
\
Rq & -{2 \ Rp}& 0 &{{{Ir \ Rq}+{2 \ Ip \ Rp}}\over Iq}
\
-{2 \ Ip}& -{2 \ Ir}&{{{2 \ Ir \ Rq}+{4 \ Ip \ Rp}}\over Iq}& 0
\
- Iq & 0 &{2 \ Rp}& - Ir
\
0 & -{2 \ Iq}&{2 \ Rq}&{2 \ Ip}](images/2587485081426453473-16.0px.png "\label{eq16}\left[
\begin{array}{cccc}
- Rq &{2 \ Rp}& 0 &{{-{Ir \ Rq}-{2 \ Ip \ Rp}}\over Iq}
\
Rq & -{2 \ Rp}& 0 &{{{Ir \ Rq}+{2 \ Ip \ Rp}}\over Iq}
\
-{2 \ Ip}& -{2 \ Ir}&{{{2 \ Ir \ Rq}+{4 \ Ip \ Rp}}\over Iq}& 0
\
- Iq & 0 &{2 \ Rp}& - Ir
\
0 & -{2 \ Iq}&{2 \ Rq}&{2 \ Ip}") | (16) |
Type: Matrix(Fraction(Polynomial(Integer)))
axiom
N:=nullSpace(H2)
![\label{eq17}\left[{\left[{{2 \ Rp}\over Iq}, \:{Rq \over Iq}, \: 1, \: 0 \right]}, \:{\left[ -{Ir \over Iq}, \:{Ip \over Iq}, \: 0, \: 1 \right]}\right]](images/4196285347253592765-16.0px.png "\label{eq17}\left[{\left[{{2 \ Rp}\over Iq}, \:{Rq \over Iq}, \: 1, \: 0 \right]}, \:{\left[ -{Ir \over Iq}, \:{Ip \over Iq}, \: 0, \: 1 \right]}\right]") | (17) |
Type: List(Vector(Fraction(Polynomial(Integer))))
axiom
H2*(c*N(1)+e*N(2))
![\label{eq18}\left[ 0, \: 0, \: 0, \: 0, \: 0 \right]](images/973788788681565529-16.0px.png "\label{eq18}\left[ 0, \: 0, \: 0, \: 0, \: 0 \right]") | (18) |
Type: Vector(Fraction(Polynomial(Integer)))
axiom
s2:=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]](images/7821801136718311984-16.0px.png "\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
h1:=map(x+->eval(x,s2), h)
![\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](images/1830753807997531664-16.0px.png "\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(Complex(Integer))))
axiom
map(x+->eval(x,s1), htranspose(ρ)*h1-h1*ρ)
![\label{eq21}\left[
\begin{array}{cc}
0 & 0
\
0 & 0](images/1510147525893569053-16.0px.png "\label{eq21}\left[
\begin{array}{cc}
0 & 0
\
0 & 0") | (21) |
Type: Matrix(Fraction(Polynomial(Complex(Integer))))