Preserve this form:
u^2+t^2-x^2-y^2
t^2-x^2
{t,x}
In[]:=
RotationMatrix[Iθ]
In[]:=
{{Cosh[θ],-Sinh[θ]},{-Sinh[θ],Cosh[θ]}}.{t,x}
Out[]=
{tCosh[θ]-xSinh[θ],xCosh[θ]-tSinh[θ]}
In[]:=
%.{{1,0},{0,-1}}.%
Out[]=
(xCosh[θ]-tSinh[θ])(-xCosh[θ]+tSinh[θ])+
2
(tCosh[θ]-xSinh[θ])
In[]:=
FullSimplify[%]
Out[]=
(t-x)(t+x)
What is matrix representation
What is matrix representation
u^2+t^2-x^2-y^2
Has 2 rotation subgroups
In[]:=
liealgebra={SparseArray[{{1,2}1,{2,1}-1},{4,4}],SparseArray[{{3,4}1,{4,3}-1},{4,4}],SparseArray[{{1,3}1,{3,1}1},{4,4}],SparseArray[{{1,4}1,{4,1}1},{4,4}],SparseArray[{{2,3}1,{3,2}1},{4,4}],SparseArray[{{2,4}1,{4,2}1},{4,4}]}
Out[]=
In[]:=
DiagonalMatrix[{1,1,-1,-1}]
Out[]=
{{1,0,0,0},{0,1,0,0},{0,0,-1,0},{0,0,0,-1}}
In[]:=
anticommute[m_]:=m.DiagonalMatrix[{1,1,-1,-1}]-DiagonalMatrix[{1,1,-1,-1}].Transpose[m]
In[]:=
anticommute/@liealgebra
Out[]=
{True,True,True,True,True,True}
In[]:=
Reduce[anticommute[Array[a,{4,4}]]]
Out[]=
a[1,1]0&&a[1,2]-a[2,1]&&a[2,2]0&&a[1,3]a[3,1]&&a[2,3]a[3,2]&&a[3,3]0&&a[1,4]a[4,1]&&a[2,4]a[4,2]&&a[3,4]-a[4,3]&&a[4,4]0
The group O(p, q) is defined for vector spaces over the reals. For complex spaces, all groups O(p, q; C) are isomorphic to the usual orthogonal group O(p + q; C), since the transform {\displaystyle z_{j}\mapsto iz_{j}}z_j \mapsto iz_j changes the signature of a form.
In[]:=
MatrixExp[t#]&/@liealgebra
Out[]=
In[]:=
#.{u,t,x,y}&/@%202//FullSimplify
Out[]=
In[]:=
#.DiagonalMatrix[{1,1,-1,-1}].#&/@%
Out[]=
In[]:=
FullSimplify[%]
Out[]=
{+--,+--,+--,+--,+--,+--}
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In[]:=
Counts[%]
Out[]=
+--6
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