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last edited 7 years ago by Bill Page |
Edit detail for SandBox Idempotent Observers revision 19 of 29
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Editor: Bill Page
Time: 2014/01/24 14:57:31 GMT+0 |
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| Note: triple products | ||
added: Is?(SS*TT*SS = dot(S,T)^2*SS) changed: -BB:=tensor(-B,B) -CC:=tensor(-C,C) BB:=tensor(-B,B); CC:=tensor(-C,C); changed: -map(Simplify,(map(possible,AA*BB*SS-SS*AA*BB))) Is?(AA*BB*CC = -dot(A,B)*dot(B,C)/dot(A,C)*AA*CC) Is?(AA*BB*SS = -dot(A,B)*dot(B,S)/dot(A,S)*AA*SS) Is?(AA*BB*SS=SS*AA*BB) Is?(AA*BB*AA = dot(A,B)^2*AA)
Preliminaries
Lorentz Form (metric) applied to a vector (
matrix)
produces a co-vector (
matrix). Scalar and tensor
products use matrix multiplication.
fricas
ID:=diagonalMatrix [1,1, 1, 1];
Type: Matrix(Integer)
fricas
G:=diagonalMatrix [-1,1, 1, 1]
![\label{eq1}\left[
\begin{array}{cccc}
- 1 & 0 & 0 & 0
\
0 & 1 & 0 & 0
\
0 & 0 & 1 & 0
\
0 & 0 & 0 & 1](images/520983853324401229-16.0px.png "\label{eq1}\left[
\begin{array}{cccc}
- 1 & 0 & 0 & 0
\
0 & 1 & 0 & 0
\
0 & 0 & 1 & 0
\
0 & 0 & 0 & 1") | (1) |
Type: Matrix(Integer)
fricas
Scalar := Expression Integer
 | (2) |
Type: Type
fricas
vect(x:List Scalar):Matrix Scalar == matrix map(y+->[y],x)
Function declaration vect : List(Expression(Integer)) -> Matrix( Expression(Integer)) has been added to workspace.
Type: Void
fricas
g(x:Matrix Scalar):Matrix Scalar == transpose(x)*G
Function declaration g : Matrix(Expression(Integer)) -> Matrix( Expression(Integer)) has been added to workspace.
Type: Void
fricas
dot(x:Matrix Scalar,y:Matrix Scalar):Scalar == g(x)*y
Function declaration dot : (Matrix(Expression(Integer)),Matrix( Expression(Integer))) -> Expression(Integer) has been added to workspace.
Type: Void
fricas
tensor(x:Matrix Scalar,y:Matrix Scalar):Matrix Scalar == x*g(y)
Function declaration tensor : (Matrix(Expression(Integer)),Matrix( Expression(Integer))) -> Matrix(Expression(Integer)) has been added to workspace.
Type: Void
Verification
fricas
htrigs2exp == rule cosh(a) == (exp(a)+exp(-a))/2 sinh(a) == (exp(a)-exp(-a))/2
Type: Void
fricas
sinhcosh == rule ?c*exp(a)+?c*exp(-a) == 2*c*cosh(a) ?c*exp(a)-?c*exp(-a) == 2*c*sinh(a) ?c*exp(a-b)+?c*exp(b-a) == 2*c*cosh(a-b) ?c*exp(a-b)-?c*exp(b-a) == 2*c*sinh(a-b)
Type: Void
fricas
expandhtrigs == rule cosh(:x+y) == sinh(x)*sinh(y)+cosh(x)*cosh(y) sinh(:x+y) == cosh(x)*sinh(y)+sinh(x)*cosh(y) cosh(2*x) == 2*cosh(x)^2-1 sinh(2*x) == 2*sinh(x)*cosh(x)
Type: Void
fricas
expandhtrigs2 == rule cosh(2*x+2*y) == 2*cosh(x+y)^2-1 sinh(2*x+2*y) == 2*sinh(x+y)*cosh(x+y) cosh(2*x-2*y) == 2*cosh(x-y)^2-1 sinh(2*x-2*y) == 2*sinh(x-y)*cosh(x-y)
Type: Void
fricas
Simplify(x:Scalar):Scalar == htrigs sinhcosh simplify htrigs2exp x
Function declaration Simplify : Expression(Integer) -> Expression( Integer) has been added to workspace.
Type: Void
fricas
possible(x)==subst(x,map(y+->(y=(random(100) - random(100))), variables x) )
Type: Void
fricas
is?(eq:Equation Scalar):Boolean == (Simplify(lhs(eq)-rhs(eq))=0)::Boolean
Function declaration is? : Equation(Expression(Integer)) -> Boolean has been added to workspace.
Type: Void
fricas
Is?(eq:Equation(Matrix(Scalar))):Boolean == _ (map(Simplify,lhs(eq)-rhs(eq)) :: Matrix Expression AlgebraicNumber = _ zero(nrows(lhs(eq)), ncols(lhs(eq)))$Matrix Expression AlgebraicNumber )::Boolean
Function declaration Is? : Equation(Matrix(Expression(Integer))) -> Boolean has been added to workspace.
Type: Void
Massive Objects
A material object (also referred to as an observer) is represented by a time-like 4-vector
fricas
P:=vect [p0,p1, p2, p3]
fricas
Compiling function vect with type List(Expression(Integer)) ->
Matrix(Expression(Integer))![\label{eq3}\left[
\begin{array}{c}
p 0
\
p 1
\
p 2
\
p 3](images/6634563813769139892-16.0px.png "\label{eq3}\left[
\begin{array}{c}
p 0
\
p 1
\
p 2
\
p 3") | (3) |
Type: Matrix(Expression(Integer))
fricas
dot(P,P)
fricas
Compiling function g with type Matrix(Expression(Integer)) -> Matrix
(Expression(Integer))fricas
Compiling function dot with type (Matrix(Expression(Integer)),Matrix (Expression(Integer))) -> Expression(Integer)
 | (4) |
Type: Expression(Integer)
fricas
solve(%=-1,p0)
![\label{eq5}\left[{p 0 ={\sqrt{{{p 3}^{2}}+{{p 2}^{2}}+{{p 1}^{2}}+ 1}}}, \:{p 0 = -{\sqrt{{{p 3}^{2}}+{{p 2}^{2}}+{{p 1}^{2}}+ 1}}}\right]](images/686934914965058360-16.0px.png "\label{eq5}\left[{p 0 ={\sqrt{{{p 3}^{2}}+{{p 2}^{2}}+{{p 1}^{2}}+ 1}}}, \:{p 0 = -{\sqrt{{{p 3}^{2}}+{{p 2}^{2}}+{{p 1}^{2}}+ 1}}}\right]") | (5) |
Type: List(Equation(Expression(Integer)))
fricas
Q:=vect [q0,q1, q2, q3];
Type: Matrix(Expression(Integer))
fricas
S:=1/sqrt(1-s1^2-s2^2-s3^2)*vect [1,-s1, -s2, -s3]
![\label{eq6}\left[
\begin{array}{c}
{1 \over{\sqrt{-{{s 3}^{2}}-{{s 2}^{2}}-{{s 1}^{2}}+ 1}}}
\
-{s 1 \over{\sqrt{-{{s 3}^{2}}-{{s 2}^{2}}-{{s 1}^{2}}+ 1}}}
\
-{s 2 \over{\sqrt{-{{s 3}^{2}}-{{s 2}^{2}}-{{s 1}^{2}}+ 1}}}
\
-{s 3 \over{\sqrt{-{{s 3}^{2}}-{{s 2}^{2}}-{{s 1}^{2}}+ 1}}}](images/9117993466847850561-16.0px.png "\label{eq6}\left[
\begin{array}{c}
{1 \over{\sqrt{-{{s 3}^{2}}-{{s 2}^{2}}-{{s 1}^{2}}+ 1}}}
\
-{s 1 \over{\sqrt{-{{s 3}^{2}}-{{s 2}^{2}}-{{s 1}^{2}}+ 1}}}
\
-{s 2 \over{\sqrt{-{{s 3}^{2}}-{{s 2}^{2}}-{{s 1}^{2}}+ 1}}}
\
-{s 3 \over{\sqrt{-{{s 3}^{2}}-{{s 2}^{2}}-{{s 1}^{2}}+ 1}}}") | (6) |
Type: Matrix(Expression(Integer))
fricas
dot(S,S)
 | (7) |
Type: Expression(Integer)
fricas
T:=1/sqrt(1-t1^2-t2^2-t3^2)*vect [1,-t1, -t2, -t3];
Type: Matrix(Expression(Integer))
fricas
W:=1/sqrt(1-w1^2-w2^2-w3^2)*vect [1,-w1, -w2, -w3];
Type: Matrix(Expression(Integer))
fricas
U:=vect [cosh(u),sinh(u), 0, 0]
![\label{eq8}\left[
\begin{array}{c}
{\cosh \left({u}\right)}
\
{\sinh \left({u}\right)}
\
0
\
0](images/4138161764451721190-16.0px.png "\label{eq8}\left[
\begin{array}{c}
{\cosh \left({u}\right)}
\
{\sinh \left({u}\right)}
\
0
\
0") | (8) |
Type: Matrix(Expression(Integer))
fricas
simplify dot(U,U)
 | (9) |
Type: Expression(Integer)
fricas
V:=vect [cosh(v),sinh(v), 0, 0];
Type: Matrix(Expression(Integer))
Massless Photons
A photon is a represented by a light-like null 4-vector
fricas
A:=vect [a0,a0*a1, a0*a2, a0*a3]
![\label{eq10}\left[
\begin{array}{c}
a 0
\
{a 0 \ a 1}
\
{a 0 \ a 2}
\
{a 0 \ a 3}](images/5432632282756795416-16.0px.png "\label{eq10}\left[
\begin{array}{c}
a 0
\
{a 0 \ a 1}
\
{a 0 \ a 2}
\
{a 0 \ a 3}") | (10) |
Type: Matrix(Expression(Integer))
fricas
solve(dot(A,A)=0, a3)
![\label{eq11}\left[{a 3 ={\sqrt{-{{a 2}^{2}}-{{a 1}^{2}}+ 1}}}, \:{a 3 = -{\sqrt{-{{a 2}^{2}}-{{a 1}^{2}}+ 1}}}\right]](images/2743663199028456662-16.0px.png "\label{eq11}\left[{a 3 ={\sqrt{-{{a 2}^{2}}-{{a 1}^{2}}+ 1}}}, \:{a 3 = -{\sqrt{-{{a 2}^{2}}-{{a 1}^{2}}+ 1}}}\right]") | (11) |
Type: List(Equation(Expression(Integer)))
fricas
A:=vect [a0,a0*a1, a0*a2, a0*sqrt(1-a1^2-a2^2)]
![\label{eq12}\left[
\begin{array}{c}
a 0
\
{a 0 \ a 1}
\
{a 0 \ a 2}
\
{a 0 \ {\sqrt{-{{a 2}^{2}}-{{a 1}^{2}}+ 1}}}](images/5594753343059314301-16.0px.png "\label{eq12}\left[
\begin{array}{c}
a 0
\
{a 0 \ a 1}
\
{a 0 \ a 2}
\
{a 0 \ {\sqrt{-{{a 2}^{2}}-{{a 1}^{2}}+ 1}}}") | (12) |
Type: Matrix(Expression(Integer))
fricas
dot(A,A)
 | (13) |
Type: Expression(Integer)
fricas
B:=vect [b0,b0*b1, b0*b2, b0*sqrt(1-b1^2-b2^2)]
![\label{eq14}\left[
\begin{array}{c}
b 0
\
{b 0 \ b 1}
\
{b 0 \ b 2}
\
{b 0 \ {\sqrt{-{{b 2}^{2}}-{{b 1}^{2}}+ 1}}}](images/8157176558800252629-16.0px.png "\label{eq14}\left[
\begin{array}{c}
b 0
\
{b 0 \ b 1}
\
{b 0 \ b 2}
\
{b 0 \ {\sqrt{-{{b 2}^{2}}-{{b 1}^{2}}+ 1}}}") | (14) |
Type: Matrix(Expression(Integer))
fricas
C:=vect [c0,c0*c1, c0*c2, c0*sqrt(1-c1^2-c2^2)]
![\label{eq15}\left[
\begin{array}{c}
c 0
\
{c 0 \ c 1}
\
{c 0 \ c 2}
\
{c 0 \ {\sqrt{-{{c 2}^{2}}-{{c 1}^{2}}+ 1}}}](images/3636827162889371438-16.0px.png "\label{eq15}\left[
\begin{array}{c}
c 0
\
{c 0 \ c 1}
\
{c 0 \ c 2}
\
{c 0 \ {\sqrt{-{{c 2}^{2}}-{{c 1}^{2}}+ 1}}}") | (15) |
Type: Matrix(Expression(Integer))
Observer "at rest"
fricas
R:=vect [1,0, 0, 0]
![\label{eq16}\left[
\begin{array}{c}
1
\
0
\
0
\
0](images/2783283433616676020-16.0px.png "\label{eq16}\left[
\begin{array}{c}
1
\
0
\
0
\
0") | (16) |
Type: Matrix(Expression(Integer))
fricas
dot(R,R)
 | (17) |
Type: Expression(Integer)
Associated with each such vector is the orthogonal 3-d Euclidean subspace
Relative Velocity
An object P has a unique relative velocity ω(P,Q) with respect to object Q given by
fricas
ω(P,Q)==-P/dot(P, Q)-Q
Type: Void
fricas
map(Simplify,ω(P, Q))
fricas
Compiling function ω with type (Matrix(Expression(Integer)),Matrix( Expression(Integer))) -> Matrix(Expression(Integer))
fricas
Compiling body of rule htrigs2exp to compute value of type Ruleset(
Integer,fricas
Compiling body of rule sinhcosh to compute value of type Ruleset(
Integer,fricas
Compiling function Simplify with type Expression(Integer) ->
Expression(Integer)![\label{eq18}\left[
\begin{array}{c}
{{-{p 3 \ q 0 \ q 3}-{p 2 \ q 0 \ q 2}-{p 1 \ q 0 \ q 1}+{p 0 \ {{q 0}^{2}}}- p 0}\over{{p 3 \ q 3}+{p 2 \ q 2}+{p 1 \ q 1}-{p 0 \ q 0}}}
\
{{-{p 3 \ q 1 \ q 3}-{p 2 \ q 1 \ q 2}-{p 1 \ {{q 1}^{2}}}+{p 0 \ q 0 \ q 1}- p 1}\over{{p 3 \ q 3}+{p 2 \ q 2}+{p 1 \ q 1}-{p 0 \ q 0}}}
\
{{-{p 3 \ q 2 \ q 3}-{p 2 \ {{q 2}^{2}}}+{{\left(-{p 1 \ q 1}+{p 0 \ q 0}\right)}\ q 2}- p 2}\over{{p 3 \ q 3}+{p 2 \ q 2}+{p 1 \ q 1}-{p 0 \ q 0}}}
\
{{-{p 3 \ {{q 3}^{2}}}+{{\left(-{p 2 \ q 2}-{p 1 \ q 1}+{p 0 \ q 0}\right)}\ q 3}- p 3}\over{{p 3 \ q 3}+{p 2 \ q 2}+{p 1 \ q 1}-{p 0 \ q 0}}}](images/4063709978803104077-16.0px.png "\label{eq18}\left[
\begin{array}{c}
{{-{p 3 \ q 0 \ q 3}-{p 2 \ q 0 \ q 2}-{p 1 \ q 0 \ q 1}+{p 0 \ {{q 0}^{2}}}- p 0}\over{{p 3 \ q 3}+{p 2 \ q 2}+{p 1 \ q 1}-{p 0 \ q 0}}}
\
{{-{p 3 \ q 1 \ q 3}-{p 2 \ q 1 \ q 2}-{p 1 \ {{q 1}^{2}}}+{p 0 \ q 0 \ q 1}- p 1}\over{{p 3 \ q 3}+{p 2 \ q 2}+{p 1 \ q 1}-{p 0 \ q 0}}}
\
{{-{p 3 \ q 2 \ q 3}-{p 2 \ {{q 2}^{2}}}+{{\left(-{p 1 \ q 1}+{p 0 \ q 0}\right)}\ q 2}- p 2}\over{{p 3 \ q 3}+{p 2 \ q 2}+{p 1 \ q 1}-{p 0 \ q 0}}}
\
{{-{p 3 \ {{q 3}^{2}}}+{{\left(-{p 2 \ q 2}-{p 1 \ q 1}+{p 0 \ q 0}\right)}\ q 3}- p 3}\over{{p 3 \ q 3}+{p 2 \ q 2}+{p 1 \ q 1}-{p 0 \ q 0}}}") | (18) |
Type: Matrix(Expression(Integer))
fricas
map(Simplify,ω(P, R))
![\label{eq19}\left[
\begin{array}{c}
0
\
{p 1 \over p 0}
\
{p 2 \over p 0}
\
{p 3 \over p 0}](images/6402536354751094971-16.0px.png "\label{eq19}\left[
\begin{array}{c}
0
\
{p 1 \over p 0}
\
{p 2 \over p 0}
\
{p 3 \over p 0}") | (19) |
Type: Matrix(Expression(Integer))
fricas
map(Simplify,ω(S, T))
![\label{eq20}\left[
\begin{array}{c}
{{{{t 3}^{2}}-{s 3 \ t 3}+{{t 2}^{2}}-{s 2 \ t 2}+{{t 1}^{2}}-{s 1 \ t 1}}\over{{\left({s 3 \ t 3}+{s 2 \ t 2}+{s 1 \ t 1}- 1 \right)}\ {\sqrt{-{{t 3}^{2}}-{{t 2}^{2}}-{{t 1}^{2}}+ 1}}}}
\
{{-{s 1 \ {{t 3}^{2}}}+{s 3 \ t 1 \ t 3}-{s 1 \ {{t 2}^{2}}}+{s 2 \ t 1 \ t 2}- t 1 + s 1}\over{{\left({s 3 \ t 3}+{s 2 \ t 2}+{s 1 \ t 1}- 1 \right)}\ {\sqrt{-{{t 3}^{2}}-{{t 2}^{2}}-{{t 1}^{2}}+ 1}}}}
\
{{-{s 2 \ {{t 3}^{2}}}+{s 3 \ t 2 \ t 3}+{{\left({s 1 \ t 1}- 1 \right)}\ t 2}-{s 2 \ {{t 1}^{2}}}+ s 2}\over{{\left({s 3 \ t 3}+{s 2 \ t 2}+{s 1 \ t 1}- 1 \right)}\ {\sqrt{-{{t 3}^{2}}-{{t 2}^{2}}-{{t 1}^{2}}+ 1}}}}
\
{{{{\left({s 2 \ t 2}+{s 1 \ t 1}- 1 \right)}\ t 3}-{s 3 \ {{t 2}^{2}}}-{s 3 \ {{t 1}^{2}}}+ s 3}\over{{\left({s 3 \ t 3}+{s 2 \ t 2}+{s 1 \ t 1}- 1 \right)}\ {\sqrt{-{{t 3}^{2}}-{{t 2}^{2}}-{{t 1}^{2}}+ 1}}}}](images/8331760300205224683-16.0px.png "\label{eq20}\left[
\begin{array}{c}
{{{{t 3}^{2}}-{s 3 \ t 3}+{{t 2}^{2}}-{s 2 \ t 2}+{{t 1}^{2}}-{s 1 \ t 1}}\over{{\left({s 3 \ t 3}+{s 2 \ t 2}+{s 1 \ t 1}- 1 \right)}\ {\sqrt{-{{t 3}^{2}}-{{t 2}^{2}}-{{t 1}^{2}}+ 1}}}}
\
{{-{s 1 \ {{t 3}^{2}}}+{s 3 \ t 1 \ t 3}-{s 1 \ {{t 2}^{2}}}+{s 2 \ t 1 \ t 2}- t 1 + s 1}\over{{\left({s 3 \ t 3}+{s 2 \ t 2}+{s 1 \ t 1}- 1 \right)}\ {\sqrt{-{{t 3}^{2}}-{{t 2}^{2}}-{{t 1}^{2}}+ 1}}}}
\
{{-{s 2 \ {{t 3}^{2}}}+{s 3 \ t 2 \ t 3}+{{\left({s 1 \ t 1}- 1 \right)}\ t 2}-{s 2 \ {{t 1}^{2}}}+ s 2}\over{{\left({s 3 \ t 3}+{s 2 \ t 2}+{s 1 \ t 1}- 1 \right)}\ {\sqrt{-{{t 3}^{2}}-{{t 2}^{2}}-{{t 1}^{2}}+ 1}}}}
\
{{{{\left({s 2 \ t 2}+{s 1 \ t 1}- 1 \right)}\ t 3}-{s 3 \ {{t 2}^{2}}}-{s 3 \ {{t 1}^{2}}}+ s 3}\over{{\left({s 3 \ t 3}+{s 2 \ t 2}+{s 1 \ t 1}- 1 \right)}\ {\sqrt{-{{t 3}^{2}}-{{t 2}^{2}}-{{t 1}^{2}}+ 1}}}}") | (20) |
Type: Matrix(Expression(Integer))
fricas
map(Simplify,ω(S, R))
![\label{eq21}\left[
\begin{array}{c}
0
\
- s 1
\
- s 2
\
- s 3](images/4044408630555344873-16.0px.png "\label{eq21}\left[
\begin{array}{c}
0
\
- s 1
\
- s 2
\
- s 3") | (21) |
Type: Matrix(Expression(Integer))
fricas
map(Simplify,ω(U, V))
![\label{eq22}\left[
\begin{array}{c}
{{-{\cosh \left({{2 \ v}- u}\right)}+{\cosh \left({u}\right)}}\over{2 \ {\cosh \left({v - u}\right)}}}
\
{{-{\sinh \left({{2 \ v}- u}\right)}+{\sinh \left({u}\right)}}\over{2 \ {\cosh \left({v - u}\right)}}}
\
0
\
0](images/928211311644225266-16.0px.png "\label{eq22}\left[
\begin{array}{c}
{{-{\cosh \left({{2 \ v}- u}\right)}+{\cosh \left({u}\right)}}\over{2 \ {\cosh \left({v - u}\right)}}}
\
{{-{\sinh \left({{2 \ v}- u}\right)}+{\sinh \left({u}\right)}}\over{2 \ {\cosh \left({v - u}\right)}}}
\
0
\
0") | (22) |
Type: Matrix(Expression(Integer))
Idempotent Observers
fricas
PP:=tensor(-P,P)
fricas
Compiling function tensor with type (Matrix(Expression(Integer)),Matrix(Expression(Integer))) -> Matrix(Expression(Integer))
![\label{eq23}\left[
\begin{array}{cccc}
{{p 0}^{2}}& -{p 0 \ p 1}& -{p 0 \ p 2}& -{p 0 \ p 3}
\
{p 0 \ p 1}& -{{p 1}^{2}}& -{p 1 \ p 2}& -{p 1 \ p 3}
\
{p 0 \ p 2}& -{p 1 \ p 2}& -{{p 2}^{2}}& -{p 2 \ p 3}
\
{p 0 \ p 3}& -{p 1 \ p 3}& -{p 2 \ p 3}& -{{p 3}^{2}}](images/1006668425770057063-16.0px.png "\label{eq23}\left[
\begin{array}{cccc}
{{p 0}^{2}}& -{p 0 \ p 1}& -{p 0 \ p 2}& -{p 0 \ p 3}
\
{p 0 \ p 1}& -{{p 1}^{2}}& -{p 1 \ p 2}& -{p 1 \ p 3}
\
{p 0 \ p 2}& -{p 1 \ p 2}& -{{p 2}^{2}}& -{p 2 \ p 3}
\
{p 0 \ p 3}& -{p 1 \ p 3}& -{p 2 \ p 3}& -{{p 3}^{2}}") | (23) |
Type: Matrix(Expression(Integer))
fricas
QQ:=tensor(-Q,Q);
Type: Matrix(Expression(Integer))
fricas
is?(trace(PP*QQ)=dot(P,Q)^2)
fricas
Compiling function is? with type Equation(Expression(Integer)) ->
Boolean | (24) |
Type: Boolean
fricas
RR:=tensor(-R,R)
![\label{eq25}\left[
\begin{array}{cccc}
1 & 0 & 0 & 0
\
0 & 0 & 0 & 0
\
0 & 0 & 0 & 0
\
0 & 0 & 0 & 0](images/5771482551771060674-16.0px.png "\label{eq25}\left[
\begin{array}{cccc}
1 & 0 & 0 & 0
\
0 & 0 & 0 & 0
\
0 & 0 & 0 & 0
\
0 & 0 & 0 & 0") | (25) |
Type: Matrix(Expression(Integer))
fricas
SS:=map(Simplify,tensor(-S, S))
![\label{eq26}\left[
\begin{array}{cccc}
-{1 \over{{{s 3}^{2}}+{{s 2}^{2}}+{{s 1}^{2}}- 1}}& -{s 1 \over{{{s 3}^{2}}+{{s 2}^{2}}+{{s 1}^{2}}- 1}}& -{s 2 \over{{{s 3}^{2}}+{{s 2}^{2}}+{{s 1}^{2}}- 1}}& -{s 3 \over{{{s 3}^{2}}+{{s 2}^{2}}+{{s 1}^{2}}- 1}}
\
{s 1 \over{{{s 3}^{2}}+{{s 2}^{2}}+{{s 1}^{2}}- 1}}&{{{s 1}^{2}}\over{{{s 3}^{2}}+{{s 2}^{2}}+{{s 1}^{2}}- 1}}&{{s 1 \ s 2}\over{{{s 3}^{2}}+{{s 2}^{2}}+{{s 1}^{2}}- 1}}&{{s 1 \ s 3}\over{{{s 3}^{2}}+{{s 2}^{2}}+{{s 1}^{2}}- 1}}
\
{s 2 \over{{{s 3}^{2}}+{{s 2}^{2}}+{{s 1}^{2}}- 1}}&{{s 1 \ s 2}\over{{{s 3}^{2}}+{{s 2}^{2}}+{{s 1}^{2}}- 1}}&{{{s 2}^{2}}\over{{{s 3}^{2}}+{{s 2}^{2}}+{{s 1}^{2}}- 1}}&{{s 2 \ s 3}\over{{{s 3}^{2}}+{{s 2}^{2}}+{{s 1}^{2}}- 1}}
\
{s 3 \over{{{s 3}^{2}}+{{s 2}^{2}}+{{s 1}^{2}}- 1}}&{{s 1 \ s 3}\over{{{s 3}^{2}}+{{s 2}^{2}}+{{s 1}^{2}}- 1}}&{{s 2 \ s 3}\over{{{s 3}^{2}}+{{s 2}^{2}}+{{s 1}^{2}}- 1}}&{{{s 3}^{2}}\over{{{s 3}^{2}}+{{s 2}^{2}}+{{s 1}^{2}}- 1}}](images/716618236896931860-16.0px.png "\label{eq26}\left[
\begin{array}{cccc}
-{1 \over{{{s 3}^{2}}+{{s 2}^{2}}+{{s 1}^{2}}- 1}}& -{s 1 \over{{{s 3}^{2}}+{{s 2}^{2}}+{{s 1}^{2}}- 1}}& -{s 2 \over{{{s 3}^{2}}+{{s 2}^{2}}+{{s 1}^{2}}- 1}}& -{s 3 \over{{{s 3}^{2}}+{{s 2}^{2}}+{{s 1}^{2}}- 1}}
\
{s 1 \over{{{s 3}^{2}}+{{s 2}^{2}}+{{s 1}^{2}}- 1}}&{{{s 1}^{2}}\over{{{s 3}^{2}}+{{s 2}^{2}}+{{s 1}^{2}}- 1}}&{{s 1 \ s 2}\over{{{s 3}^{2}}+{{s 2}^{2}}+{{s 1}^{2}}- 1}}&{{s 1 \ s 3}\over{{{s 3}^{2}}+{{s 2}^{2}}+{{s 1}^{2}}- 1}}
\
{s 2 \over{{{s 3}^{2}}+{{s 2}^{2}}+{{s 1}^{2}}- 1}}&{{s 1 \ s 2}\over{{{s 3}^{2}}+{{s 2}^{2}}+{{s 1}^{2}}- 1}}&{{{s 2}^{2}}\over{{{s 3}^{2}}+{{s 2}^{2}}+{{s 1}^{2}}- 1}}&{{s 2 \ s 3}\over{{{s 3}^{2}}+{{s 2}^{2}}+{{s 1}^{2}}- 1}}
\
{s 3 \over{{{s 3}^{2}}+{{s 2}^{2}}+{{s 1}^{2}}- 1}}&{{s 1 \ s 3}\over{{{s 3}^{2}}+{{s 2}^{2}}+{{s 1}^{2}}- 1}}&{{s 2 \ s 3}\over{{{s 3}^{2}}+{{s 2}^{2}}+{{s 1}^{2}}- 1}}&{{{s 3}^{2}}\over{{{s 3}^{2}}+{{s 2}^{2}}+{{s 1}^{2}}- 1}}") | (26) |
Type: Matrix(Expression(Integer))
fricas
Is?(SS*SS=SS)
fricas
Compiling function Is? with type Equation(Matrix(Expression(Integer)
)) -> Boolean | (27) |
Type: Boolean
fricas
trace(SS)
 | (28) |
Type: Expression(Integer)
fricas
TT:=map(Simplify,tensor(-T, T));
Type: Matrix(Expression(Integer))
fricas
Is?(SS*TT*SS = dot(S,T)^2 * SS)
 | (29) |
Type: Boolean
fricas
UU:=map(Simplify,tensor(-U, U))
![\label{eq30}\left[
\begin{array}{cccc}
{{{\cosh \left({2 \ u}\right)}+ 1}\over 2}& -{{\sinh \left({2 \ u}\right)}\over 2}& 0 & 0
\
{{\sinh \left({2 \ u}\right)}\over 2}&{{-{\cosh \left({2 \ u}\right)}+ 1}\over 2}& 0 & 0
\
0 & 0 & 0 & 0
\
0 & 0 & 0 & 0](images/8726807008798667260-16.0px.png "\label{eq30}\left[
\begin{array}{cccc}
{{{\cosh \left({2 \ u}\right)}+ 1}\over 2}& -{{\sinh \left({2 \ u}\right)}\over 2}& 0 & 0
\
{{\sinh \left({2 \ u}\right)}\over 2}&{{-{\cosh \left({2 \ u}\right)}+ 1}\over 2}& 0 & 0
\
0 & 0 & 0 & 0
\
0 & 0 & 0 & 0") | (30) |
Type: Matrix(Expression(Integer))
fricas
VV:=map(Simplify,tensor(-V, V));
Type: Matrix(Expression(Integer))
fricas
map(Simplify,UU*VV)
![\label{eq31}\left[
\begin{array}{cccc}
{{{\cosh \left({2 \ v}\right)}+{\cosh \left({{2 \ v}-{2 \ u}}\right)}+{\cosh \left({2 \ u}\right)}+ 1}\over 4}&{{-{\sinh \left({2 \ v}\right)}-{\sinh \left({{2 \ v}-{2 \ u}}\right)}-{\sinh \left({2 \ u}\right)}}\over 4}& 0 & 0
\
{{{\sinh \left({2 \ v}\right)}-{\sinh \left({{2 \ v}-{2 \ u}}\right)}+{\sinh \left({2 \ u}\right)}}\over 4}&{{-{\cosh \left({2 \ v}\right)}+{\cosh \left({{2 \ v}-{2 \ u}}\right)}-{\cosh \left({2 \ u}\right)}+ 1}\over 4}& 0 & 0
\
0 & 0 & 0 & 0
\
0 & 0 & 0 & 0](images/1326990752696507572-16.0px.png "\label{eq31}\left[
\begin{array}{cccc}
{{{\cosh \left({2 \ v}\right)}+{\cosh \left({{2 \ v}-{2 \ u}}\right)}+{\cosh \left({2 \ u}\right)}+ 1}\over 4}&{{-{\sinh \left({2 \ v}\right)}-{\sinh \left({{2 \ v}-{2 \ u}}\right)}-{\sinh \left({2 \ u}\right)}}\over 4}& 0 & 0
\
{{{\sinh \left({2 \ v}\right)}-{\sinh \left({{2 \ v}-{2 \ u}}\right)}+{\sinh \left({2 \ u}\right)}}\over 4}&{{-{\cosh \left({2 \ v}\right)}+{\cosh \left({{2 \ v}-{2 \ u}}\right)}-{\cosh \left({2 \ u}\right)}+ 1}\over 4}& 0 & 0
\
0 & 0 & 0 & 0
\
0 & 0 & 0 & 0") | (31) |
Type: Matrix(Expression(Integer))
fricas
WW:=map(Simplify,tensor(-W, W));
Type: Matrix(Expression(Integer))
fricas
Is?(SS*TT*WW = -dot(S,T)*dot(T, W)/dot(S, W)*SS*WW)
 | (32) |
Type: Boolean
fricas
Is?(SS*TT*SS = dot(S,T)^2*SS)
 | (33) |
Type: Boolean
Nilpotent Operators
fricas
AA:=tensor(-A,A)
![\label{eq34}\left[
\begin{array}{cccc}
{{a 0}^{2}}& -{{{a 0}^{2}}\ a 1}& -{{{a 0}^{2}}\ a 2}& -{{{a 0}^{2}}\ {\sqrt{-{{a 2}^{2}}-{{a 1}^{2}}+ 1}}}
\
{{{a 0}^{2}}\ a 1}& -{{{a 0}^{2}}\ {{a 1}^{2}}}& -{{{a 0}^{2}}\ a 1 \ a 2}& -{{{a 0}^{2}}\ a 1 \ {\sqrt{-{{a 2}^{2}}-{{a 1}^{2}}+ 1}}}
\
{{{a 0}^{2}}\ a 2}& -{{{a 0}^{2}}\ a 1 \ a 2}& -{{{a 0}^{2}}\ {{a 2}^{2}}}& -{{{a 0}^{2}}\ a 2 \ {\sqrt{-{{a 2}^{2}}-{{a 1}^{2}}+ 1}}}
\
{{{a 0}^{2}}\ {\sqrt{-{{a 2}^{2}}-{{a 1}^{2}}+ 1}}}& -{{{a 0}^{2}}\ a 1 \ {\sqrt{-{{a 2}^{2}}-{{a 1}^{2}}+ 1}}}& -{{{a 0}^{2}}\ a 2 \ {\sqrt{-{{a 2}^{2}}-{{a 1}^{2}}+ 1}}}&{{{{a 0}^{2}}\ {{a 2}^{2}}}+{{{a 0}^{2}}\ {{a 1}^{2}}}-{{a 0}^{2}}}](images/1244227550011891776-16.0px.png "\label{eq34}\left[
\begin{array}{cccc}
{{a 0}^{2}}& -{{{a 0}^{2}}\ a 1}& -{{{a 0}^{2}}\ a 2}& -{{{a 0}^{2}}\ {\sqrt{-{{a 2}^{2}}-{{a 1}^{2}}+ 1}}}
\
{{{a 0}^{2}}\ a 1}& -{{{a 0}^{2}}\ {{a 1}^{2}}}& -{{{a 0}^{2}}\ a 1 \ a 2}& -{{{a 0}^{2}}\ a 1 \ {\sqrt{-{{a 2}^{2}}-{{a 1}^{2}}+ 1}}}
\
{{{a 0}^{2}}\ a 2}& -{{{a 0}^{2}}\ a 1 \ a 2}& -{{{a 0}^{2}}\ {{a 2}^{2}}}& -{{{a 0}^{2}}\ a 2 \ {\sqrt{-{{a 2}^{2}}-{{a 1}^{2}}+ 1}}}
\
{{{a 0}^{2}}\ {\sqrt{-{{a 2}^{2}}-{{a 1}^{2}}+ 1}}}& -{{{a 0}^{2}}\ a 1 \ {\sqrt{-{{a 2}^{2}}-{{a 1}^{2}}+ 1}}}& -{{{a 0}^{2}}\ a 2 \ {\sqrt{-{{a 2}^{2}}-{{a 1}^{2}}+ 1}}}&{{{{a 0}^{2}}\ {{a 2}^{2}}}+{{{a 0}^{2}}\ {{a 1}^{2}}}-{{a 0}^{2}}}") | (34) |
Type: Matrix(Expression(Integer))
fricas
BB:=tensor(-B,B);
Type: Matrix(Expression(Integer))
fricas
CC:=tensor(-C,C);
Type: Matrix(Expression(Integer))
fricas
Is?(AA*AA=0*AA)
 | (35) |
Type: Boolean
fricas
trace(AA)
 | (36) |
Type: Expression(Integer)
fricas
is?(trace(AA*BB)=dot(A,B)^2)
 | (37) |
Type: Boolean
fricas
dot(A,B)
![\label{eq38}\begin{array}{@{}l}
\displaystyle
{a 0 \ b 0 \ {\sqrt{-{{b 2}^{2}}-{{b 1}^{2}}+ 1}}\ {\sqrt{-{{a 2}^{2}}-{{a 1}^{2}}+ 1}}}+{a 0 \ a 2 \ b 0 \ b 2}+
\
\
\displaystyle
{a 0 \ a 1 \ b 0 \ b 1}-{a 0 \ b 0}](images/5788218167477879858-16.0px.png "\label{eq38}\begin{array}{@{}l}
\displaystyle
{a 0 \ b 0 \ {\sqrt{-{{b 2}^{2}}-{{b 1}^{2}}+ 1}}\ {\sqrt{-{{a 2}^{2}}-{{a 1}^{2}}+ 1}}}+{a 0 \ a 2 \ b 0 \ b 2}+
\
\
\displaystyle
{a 0 \ a 1 \ b 0 \ b 1}-{a 0 \ b 0}") | (38) |
Type: Expression(Integer)
fricas
possible(%)::Complex Float
fricas
Compiling function possible with type Expression(Integer) ->
Expression(Integer) | (39) |
Type: Complex(Float)
fricas
Is?(AA*BB*CC = -dot(A,B)*dot(B, C)/dot(A, C)*AA*CC)
 | (40) |
Type: Boolean
fricas
Is?(AA*BB*SS = -dot(A,B)*dot(B, S)/dot(A, S)*AA*SS)
 | (41) |
Type: Boolean
fricas
Is?(AA*BB*SS=SS*AA*BB)
 | (42) |
Type: Boolean
fricas
Is?(AA*BB*AA = dot(A,B)^2*AA)
 | (43) |
Type: Boolean
Unit
fricas
-(PP*QQ+QQ*PP)+PP+QQ;
Type: Matrix(Expression(Integer))
fricas
2/trace(%)*%;
Type: Matrix(Expression(Integer))
fricas
trace %
 | (44) |
Type: Expression(Integer)
fricas
n:=map(Simplify,-(SS*TT+TT*SS)+SS+TT);
Type: Matrix(Expression(Integer))
fricas
Is?(n = (SS-TT)*(SS-TT) )
 | (45) |
Type: Boolean
fricas
N:=map(Simplify,2/trace(n)*n);
Type: Matrix(Expression(Integer))
fricas
Simplify trace N
 | (46) |
Type: Expression(Integer)
fricas
Is?(N*N=N)
 | (47) |
Type: Boolean
fricas
Is?(SS*N=SS)
 | (48) |
Type: Boolean
fricas
Is?(TT*N=TT)
 | (49) |
Type: Boolean
fricas
Is?(N*SS=SS)
 | (50) |
Type: Boolean
fricas
Is?(N*TT=TT)
 | (51) |
Type: Boolean
fricas
-(PP*RR+RR*PP)+PP+RR;
Type: Matrix(Expression(Integer))
fricas
2/trace(%)*%
![\label{eq52}\left[
\begin{array}{cccc}
{{{2 \ {{p 0}^{2}}}- 2}\over{{{p 3}^{2}}+{{p 2}^{2}}+{{p 1}^{2}}+{{p 0}^{2}}- 1}}& 0 & 0 & 0
\
0 &{{2 \ {{p 1}^{2}}}\over{{{p 3}^{2}}+{{p 2}^{2}}+{{p 1}^{2}}+{{p 0}^{2}}- 1}}&{{2 \ p 1 \ p 2}\over{{{p 3}^{2}}+{{p 2}^{2}}+{{p 1}^{2}}+{{p 0}^{2}}- 1}}&{{2 \ p 1 \ p 3}\over{{{p 3}^{2}}+{{p 2}^{2}}+{{p 1}^{2}}+{{p 0}^{2}}- 1}}
\
0 &{{2 \ p 1 \ p 2}\over{{{p 3}^{2}}+{{p 2}^{2}}+{{p 1}^{2}}+{{p 0}^{2}}- 1}}&{{2 \ {{p 2}^{2}}}\over{{{p 3}^{2}}+{{p 2}^{2}}+{{p 1}^{2}}+{{p 0}^{2}}- 1}}&{{2 \ p 2 \ p 3}\over{{{p 3}^{2}}+{{p 2}^{2}}+{{p 1}^{2}}+{{p 0}^{2}}- 1}}
\
0 &{{2 \ p 1 \ p 3}\over{{{p 3}^{2}}+{{p 2}^{2}}+{{p 1}^{2}}+{{p 0}^{2}}- 1}}&{{2 \ p 2 \ p 3}\over{{{p 3}^{2}}+{{p 2}^{2}}+{{p 1}^{2}}+{{p 0}^{2}}- 1}}&{{2 \ {{p 3}^{2}}}\over{{{p 3}^{2}}+{{p 2}^{2}}+{{p 1}^{2}}+{{p 0}^{2}}- 1}}](images/2679726994698647768-16.0px.png "\label{eq52}\left[
\begin{array}{cccc}
{{{2 \ {{p 0}^{2}}}- 2}\over{{{p 3}^{2}}+{{p 2}^{2}}+{{p 1}^{2}}+{{p 0}^{2}}- 1}}& 0 & 0 & 0
\
0 &{{2 \ {{p 1}^{2}}}\over{{{p 3}^{2}}+{{p 2}^{2}}+{{p 1}^{2}}+{{p 0}^{2}}- 1}}&{{2 \ p 1 \ p 2}\over{{{p 3}^{2}}+{{p 2}^{2}}+{{p 1}^{2}}+{{p 0}^{2}}- 1}}&{{2 \ p 1 \ p 3}\over{{{p 3}^{2}}+{{p 2}^{2}}+{{p 1}^{2}}+{{p 0}^{2}}- 1}}
\
0 &{{2 \ p 1 \ p 2}\over{{{p 3}^{2}}+{{p 2}^{2}}+{{p 1}^{2}}+{{p 0}^{2}}- 1}}&{{2 \ {{p 2}^{2}}}\over{{{p 3}^{2}}+{{p 2}^{2}}+{{p 1}^{2}}+{{p 0}^{2}}- 1}}&{{2 \ p 2 \ p 3}\over{{{p 3}^{2}}+{{p 2}^{2}}+{{p 1}^{2}}+{{p 0}^{2}}- 1}}
\
0 &{{2 \ p 1 \ p 3}\over{{{p 3}^{2}}+{{p 2}^{2}}+{{p 1}^{2}}+{{p 0}^{2}}- 1}}&{{2 \ p 2 \ p 3}\over{{{p 3}^{2}}+{{p 2}^{2}}+{{p 1}^{2}}+{{p 0}^{2}}- 1}}&{{2 \ {{p 3}^{2}}}\over{{{p 3}^{2}}+{{p 2}^{2}}+{{p 1}^{2}}+{{p 0}^{2}}- 1}}") | (52) |
Type: Matrix(Expression(Integer))
fricas
-(SS*RR+RR*SS)+SS+RR;
Type: Matrix(Expression(Integer))
fricas
2/trace(%)*%
![\label{eq53}\left[
\begin{array}{cccc}
1 & 0 & 0 & 0
\
0 &{{{s 1}^{2}}\over{{{s 3}^{2}}+{{s 2}^{2}}+{{s 1}^{2}}}}&{{s 1 \ s 2}\over{{{s 3}^{2}}+{{s 2}^{2}}+{{s 1}^{2}}}}&{{s 1 \ s 3}\over{{{s 3}^{2}}+{{s 2}^{2}}+{{s 1}^{2}}}}
\
0 &{{s 1 \ s 2}\over{{{s 3}^{2}}+{{s 2}^{2}}+{{s 1}^{2}}}}&{{{s 2}^{2}}\over{{{s 3}^{2}}+{{s 2}^{2}}+{{s 1}^{2}}}}&{{s 2 \ s 3}\over{{{s 3}^{2}}+{{s 2}^{2}}+{{s 1}^{2}}}}
\
0 &{{s 1 \ s 3}\over{{{s 3}^{2}}+{{s 2}^{2}}+{{s 1}^{2}}}}&{{s 2 \ s 3}\over{{{s 3}^{2}}+{{s 2}^{2}}+{{s 1}^{2}}}}&{{{s 3}^{2}}\over{{{s 3}^{2}}+{{s 2}^{2}}+{{s 1}^{2}}}}](images/7108596704844988865-16.0px.png "\label{eq53}\left[
\begin{array}{cccc}
1 & 0 & 0 & 0
\
0 &{{{s 1}^{2}}\over{{{s 3}^{2}}+{{s 2}^{2}}+{{s 1}^{2}}}}&{{s 1 \ s 2}\over{{{s 3}^{2}}+{{s 2}^{2}}+{{s 1}^{2}}}}&{{s 1 \ s 3}\over{{{s 3}^{2}}+{{s 2}^{2}}+{{s 1}^{2}}}}
\
0 &{{s 1 \ s 2}\over{{{s 3}^{2}}+{{s 2}^{2}}+{{s 1}^{2}}}}&{{{s 2}^{2}}\over{{{s 3}^{2}}+{{s 2}^{2}}+{{s 1}^{2}}}}&{{s 2 \ s 3}\over{{{s 3}^{2}}+{{s 2}^{2}}+{{s 1}^{2}}}}
\
0 &{{s 1 \ s 3}\over{{{s 3}^{2}}+{{s 2}^{2}}+{{s 1}^{2}}}}&{{s 2 \ s 3}\over{{{s 3}^{2}}+{{s 2}^{2}}+{{s 1}^{2}}}}&{{{s 3}^{2}}\over{{{s 3}^{2}}+{{s 2}^{2}}+{{s 1}^{2}}}}") | (53) |
Type: Matrix(Expression(Integer))
fricas
-(UU*RR+RR*UU)+UU+RR;
Type: Matrix(Expression(Integer))
fricas
map(Simplify,2/trace(%)*%)
![\label{eq54}\left[
\begin{array}{cccc}
1 & 0 & 0 & 0
\
0 & 1 & 0 & 0
\
0 & 0 & 0 & 0
\
0 & 0 & 0 & 0](images/8475975026758083437-16.0px.png "\label{eq54}\left[
\begin{array}{cccc}
1 & 0 & 0 & 0
\
0 & 1 & 0 & 0
\
0 & 0 & 0 & 0
\
0 & 0 & 0 & 0") | (54) |
Type: Matrix(Expression(Integer))
fricas
-(UU*VV+VV*UU)+UU+VV;
Type: Matrix(Expression(Integer))
fricas
map(Simplify,2/trace(%)*%)
![\label{eq55}\left[
\begin{array}{cccc}
1 & 0 & 0 & 0
\
0 & 1 & 0 & 0
\
0 & 0 & 0 & 0
\
0 & 0 & 0 & 0](images/8169704834814624210-16.0px.png "\label{eq55}\left[
\begin{array}{cccc}
1 & 0 & 0 & 0
\
0 & 1 & 0 & 0
\
0 & 0 & 0 & 0
\
0 & 0 & 0 & 0") | (55) |
Type: Matrix(Expression(Integer))
Momentum
fricas
m:=map(x+->factor(numer x)/factor(denom x),-(PP*QQ+QQ*PP)/dot(P, Q)+PP+QQ)
![\label{eq56}\left[
\begin{array}{cccc}
{{\left(q 0 + p 0 \right)}^{2}}& -{{\left(q 0 + p 0 \right)}\ {\left(q 1 + p 1 \right)}}& -{{\left(q 0 + p 0 \right)}\ {\left(q 2 + p 2 \right)}}& -{{\left(q 0 + p 0 \right)}\ {\left(q 3 + p 3 \right)}}
\
{{\left(q 0 + p 0 \right)}\ {\left(q 1 + p 1 \right)}}& -{{\left(q 1 + p 1 \right)}^{2}}& -{{\left(q 1 + p 1 \right)}\ {\left(q 2 + p 2 \right)}}& -{{\left(q 1 + p 1 \right)}\ {\left(q 3 + p 3 \right)}}
\
{{\left(q 0 + p 0 \right)}\ {\left(q 2 + p 2 \right)}}& -{{\left(q 1 + p 1 \right)}\ {\left(q 2 + p 2 \right)}}& -{{\left(q 2 + p 2 \right)}^{2}}& -{{\left(q 2 + p 2 \right)}\ {\left(q 3 + p 3 \right)}}
\
{{\left(q 0 + p 0 \right)}\ {\left(q 3 + p 3 \right)}}& -{{\left(q 1 + p 1 \right)}\ {\left(q 3 + p 3 \right)}}& -{{\left(q 2 + p 2 \right)}\ {\left(q 3 + p 3 \right)}}& -{{\left(q 3 + p 3 \right)}^{2}}](images/8880149671469751283-16.0px.png "\label{eq56}\left[
\begin{array}{cccc}
{{\left(q 0 + p 0 \right)}^{2}}& -{{\left(q 0 + p 0 \right)}\ {\left(q 1 + p 1 \right)}}& -{{\left(q 0 + p 0 \right)}\ {\left(q 2 + p 2 \right)}}& -{{\left(q 0 + p 0 \right)}\ {\left(q 3 + p 3 \right)}}
\
{{\left(q 0 + p 0 \right)}\ {\left(q 1 + p 1 \right)}}& -{{\left(q 1 + p 1 \right)}^{2}}& -{{\left(q 1 + p 1 \right)}\ {\left(q 2 + p 2 \right)}}& -{{\left(q 1 + p 1 \right)}\ {\left(q 3 + p 3 \right)}}
\
{{\left(q 0 + p 0 \right)}\ {\left(q 2 + p 2 \right)}}& -{{\left(q 1 + p 1 \right)}\ {\left(q 2 + p 2 \right)}}& -{{\left(q 2 + p 2 \right)}^{2}}& -{{\left(q 2 + p 2 \right)}\ {\left(q 3 + p 3 \right)}}
\
{{\left(q 0 + p 0 \right)}\ {\left(q 3 + p 3 \right)}}& -{{\left(q 1 + p 1 \right)}\ {\left(q 3 + p 3 \right)}}& -{{\left(q 2 + p 2 \right)}\ {\left(q 3 + p 3 \right)}}& -{{\left(q 3 + p 3 \right)}^{2}}") | (56) |
Type: Matrix(Fraction(Factored(SparseMultivariatePolynomial?(Integer,
fricas
Is?(m = tensor(-(P+Q),(P+Q)))
 | (57) |
Type: Boolean
fricas
factor trace m
![\label{eq58}\begin{array}{@{}l}
\displaystyle
-{\left({
\begin{array}{@{}l}
\displaystyle
{{q 3}^{2}}+{2 \ p 3 \ q 3}+{{q 2}^{2}}+{2 \ p 2 \ q 2}+{{q 1}^{2}}+{2 \ p 1 \ q 1}-
\
\
\displaystyle
{{q 0}^{2}}-{2 \ p 0 \ q 0}+{{p 3}^{2}}+{{p 2}^{2}}+{{p 1}^{2}}-{{p 0}^{2}}](images/2136842428423207941-16.0px.png "\label{eq58}\begin{array}{@{}l}
\displaystyle
-{\left({
\begin{array}{@{}l}
\displaystyle
{{q 3}^{2}}+{2 \ p 3 \ q 3}+{{q 2}^{2}}+{2 \ p 2 \ q 2}+{{q 1}^{2}}+{2 \ p 1 \ q 1}-
\
\
\displaystyle
{{q 0}^{2}}-{2 \ p 0 \ q 0}+{{p 3}^{2}}+{{p 2}^{2}}+{{p 1}^{2}}-{{p 0}^{2}}") | (58) |
Type: Factored(SparseMultivariatePolynomial?(Integer,
fricas
M:=1/trace(m)*m::Matrix Scalar;
Type: Matrix(Expression(Integer))
fricas
trace M
 | (59) |
Type: Expression(Integer)
fricas
Is?(M*M=M)
 | (60) |
Type: Boolean
fricas
(PP*RR+RR*PP)/dot(P,R)-PP-RR;
Type: Matrix(Expression(Integer))
fricas
1/trace(%)*%
![\label{eq61}\left[
\begin{array}{cccc}
{{-{{p 0}^{2}}-{2 \ p 0}- 1}\over{{{p 3}^{2}}+{{p 2}^{2}}+{{p 1}^{2}}-{{p 0}^{2}}-{2 \ p 0}- 1}}&{{{\left(p 0 + 1 \right)}\ p 1}\over{{{p 3}^{2}}+{{p 2}^{2}}+{{p 1}^{2}}-{{p 0}^{2}}-{2 \ p 0}- 1}}&{{{\left(p 0 + 1 \right)}\ p 2}\over{{{p 3}^{2}}+{{p 2}^{2}}+{{p 1}^{2}}-{{p 0}^{2}}-{2 \ p 0}- 1}}&{{{\left(p 0 + 1 \right)}\ p 3}\over{{{p 3}^{2}}+{{p 2}^{2}}+{{p 1}^{2}}-{{p 0}^{2}}-{2 \ p 0}- 1}}
\
{{{\left(- p 0 - 1 \right)}\ p 1}\over{{{p 3}^{2}}+{{p 2}^{2}}+{{p 1}^{2}}-{{p 0}^{2}}-{2 \ p 0}- 1}}&{{{p 1}^{2}}\over{{{p 3}^{2}}+{{p 2}^{2}}+{{p 1}^{2}}-{{p 0}^{2}}-{2 \ p 0}- 1}}&{{p 1 \ p 2}\over{{{p 3}^{2}}+{{p 2}^{2}}+{{p 1}^{2}}-{{p 0}^{2}}-{2 \ p 0}- 1}}&{{p 1 \ p 3}\over{{{p 3}^{2}}+{{p 2}^{2}}+{{p 1}^{2}}-{{p 0}^{2}}-{2 \ p 0}- 1}}
\
{{{\left(- p 0 - 1 \right)}\ p 2}\over{{{p 3}^{2}}+{{p 2}^{2}}+{{p 1}^{2}}-{{p 0}^{2}}-{2 \ p 0}- 1}}&{{p 1 \ p 2}\over{{{p 3}^{2}}+{{p 2}^{2}}+{{p 1}^{2}}-{{p 0}^{2}}-{2 \ p 0}- 1}}&{{{p 2}^{2}}\over{{{p 3}^{2}}+{{p 2}^{2}}+{{p 1}^{2}}-{{p 0}^{2}}-{2 \ p 0}- 1}}&{{p 2 \ p 3}\over{{{p 3}^{2}}+{{p 2}^{2}}+{{p 1}^{2}}-{{p 0}^{2}}-{2 \ p 0}- 1}}
\
{{{\left(- p 0 - 1 \right)}\ p 3}\over{{{p 3}^{2}}+{{p 2}^{2}}+{{p 1}^{2}}-{{p 0}^{2}}-{2 \ p 0}- 1}}&{{p 1 \ p 3}\over{{{p 3}^{2}}+{{p 2}^{2}}+{{p 1}^{2}}-{{p 0}^{2}}-{2 \ p 0}- 1}}&{{p 2 \ p 3}\over{{{p 3}^{2}}+{{p 2}^{2}}+{{p 1}^{2}}-{{p 0}^{2}}-{2 \ p 0}- 1}}&{{{p 3}^{2}}\over{{{p 3}^{2}}+{{p 2}^{2}}+{{p 1}^{2}}-{{p 0}^{2}}-{2 \ p 0}- 1}}](images/3149379953359539133-16.0px.png "\label{eq61}\left[
\begin{array}{cccc}
{{-{{p 0}^{2}}-{2 \ p 0}- 1}\over{{{p 3}^{2}}+{{p 2}^{2}}+{{p 1}^{2}}-{{p 0}^{2}}-{2 \ p 0}- 1}}&{{{\left(p 0 + 1 \right)}\ p 1}\over{{{p 3}^{2}}+{{p 2}^{2}}+{{p 1}^{2}}-{{p 0}^{2}}-{2 \ p 0}- 1}}&{{{\left(p 0 + 1 \right)}\ p 2}\over{{{p 3}^{2}}+{{p 2}^{2}}+{{p 1}^{2}}-{{p 0}^{2}}-{2 \ p 0}- 1}}&{{{\left(p 0 + 1 \right)}\ p 3}\over{{{p 3}^{2}}+{{p 2}^{2}}+{{p 1}^{2}}-{{p 0}^{2}}-{2 \ p 0}- 1}}
\
{{{\left(- p 0 - 1 \right)}\ p 1}\over{{{p 3}^{2}}+{{p 2}^{2}}+{{p 1}^{2}}-{{p 0}^{2}}-{2 \ p 0}- 1}}&{{{p 1}^{2}}\over{{{p 3}^{2}}+{{p 2}^{2}}+{{p 1}^{2}}-{{p 0}^{2}}-{2 \ p 0}- 1}}&{{p 1 \ p 2}\over{{{p 3}^{2}}+{{p 2}^{2}}+{{p 1}^{2}}-{{p 0}^{2}}-{2 \ p 0}- 1}}&{{p 1 \ p 3}\over{{{p 3}^{2}}+{{p 2}^{2}}+{{p 1}^{2}}-{{p 0}^{2}}-{2 \ p 0}- 1}}
\
{{{\left(- p 0 - 1 \right)}\ p 2}\over{{{p 3}^{2}}+{{p 2}^{2}}+{{p 1}^{2}}-{{p 0}^{2}}-{2 \ p 0}- 1}}&{{p 1 \ p 2}\over{{{p 3}^{2}}+{{p 2}^{2}}+{{p 1}^{2}}-{{p 0}^{2}}-{2 \ p 0}- 1}}&{{{p 2}^{2}}\over{{{p 3}^{2}}+{{p 2}^{2}}+{{p 1}^{2}}-{{p 0}^{2}}-{2 \ p 0}- 1}}&{{p 2 \ p 3}\over{{{p 3}^{2}}+{{p 2}^{2}}+{{p 1}^{2}}-{{p 0}^{2}}-{2 \ p 0}- 1}}
\
{{{\left(- p 0 - 1 \right)}\ p 3}\over{{{p 3}^{2}}+{{p 2}^{2}}+{{p 1}^{2}}-{{p 0}^{2}}-{2 \ p 0}- 1}}&{{p 1 \ p 3}\over{{{p 3}^{2}}+{{p 2}^{2}}+{{p 1}^{2}}-{{p 0}^{2}}-{2 \ p 0}- 1}}&{{p 2 \ p 3}\over{{{p 3}^{2}}+{{p 2}^{2}}+{{p 1}^{2}}-{{p 0}^{2}}-{2 \ p 0}- 1}}&{{{p 3}^{2}}\over{{{p 3}^{2}}+{{p 2}^{2}}+{{p 1}^{2}}-{{p 0}^{2}}-{2 \ p 0}- 1}}") | (61) |
Type: Matrix(Expression(Integer))
fricas
trace %
 | (62) |
Type: Expression(Integer)
fricas
--(SS*RR+RR*SS)/dot(S,R)-SS-RR; --map(Simplify, 1/trace(%)*%) --(UU*VV+VV*UU)/dot(U, V)-UU-VV; --1/trace(%)*% (UU*RR+RR*UU)/dot(U, R)-UU-RR;
Type: Matrix(Expression(Integer))
fricas
map(Simplify,1/trace(%)*%)
![\label{eq63}\left[
\begin{array}{cccc}
{{{\cosh \left({3 \ u}\right)}+{4 \ {\cosh \left({2 \ u}\right)}}+{7 \ {\cosh \left({u}\right)}}+ 4}\over{{4 \ {\cosh \left({2 \ u}\right)}}+{8 \ {\cosh \left({u}\right)}}+ 4}}&{{-{\sinh \left({3 \ u}\right)}-{2 \ {\sinh \left({2 \ u}\right)}}-{\sinh \left({u}\right)}}\over{{4 \ {\cosh \left({2 \ u}\right)}}+{8 \ {\cosh \left({u}\right)}}+ 4}}& 0 & 0
\
{{{\sinh \left({3 \ u}\right)}+{2 \ {\sinh \left({2 \ u}\right)}}+{\sinh \left({u}\right)}}\over{{4 \ {\cosh \left({2 \ u}\right)}}+{8 \ {\cosh \left({u}\right)}}+ 4}}&{{-{\cosh \left({3 \ u}\right)}+{\cosh \left({u}\right)}}\over{{4 \ {\cosh \left({2 \ u}\right)}}+{8 \ {\cosh \left({u}\right)}}+ 4}}& 0 & 0
\
0 & 0 & 0 & 0
\
0 & 0 & 0 & 0](images/7587200329369887365-16.0px.png "\label{eq63}\left[
\begin{array}{cccc}
{{{\cosh \left({3 \ u}\right)}+{4 \ {\cosh \left({2 \ u}\right)}}+{7 \ {\cosh \left({u}\right)}}+ 4}\over{{4 \ {\cosh \left({2 \ u}\right)}}+{8 \ {\cosh \left({u}\right)}}+ 4}}&{{-{\sinh \left({3 \ u}\right)}-{2 \ {\sinh \left({2 \ u}\right)}}-{\sinh \left({u}\right)}}\over{{4 \ {\cosh \left({2 \ u}\right)}}+{8 \ {\cosh \left({u}\right)}}+ 4}}& 0 & 0
\
{{{\sinh \left({3 \ u}\right)}+{2 \ {\sinh \left({2 \ u}\right)}}+{\sinh \left({u}\right)}}\over{{4 \ {\cosh \left({2 \ u}\right)}}+{8 \ {\cosh \left({u}\right)}}+ 4}}&{{-{\cosh \left({3 \ u}\right)}+{\cosh \left({u}\right)}}\over{{4 \ {\cosh \left({2 \ u}\right)}}+{8 \ {\cosh \left({u}\right)}}+ 4}}& 0 & 0
\
0 & 0 & 0 & 0
\
0 & 0 & 0 & 0") | (63) |
Type: Matrix(Expression(Integer))