- #1
Diophantus
- 69
- 0
Hi,
I am trying to find the tangent space of SL(n,real) where A(0) is defined to be the identity matrix.
First of all I worked on the case when n=2 and found that the tangent space was
A = \left( \begin{array}{ccc}<br /> a & b \\<br /> c & -a<br /> \end{array} \right)
where a,b,c belong to the reals,
so I made the conjecture that for n in general, the tangent space would be the space of traceless matrices.
I attempted to prove this by showing that the tangent space and the space of traceless matrices were subsets of each other. Whilst I could show that an arbitary element of the tangent space is traceless, I could not show the converse.
Do I just need to try harder or is my conjecture just plain wrong?
PS. I used the standard result: d/dt (detA(0)) = tr(dA(0)/dt)
I have reason to believe that det(exp(A)) = exp(tr(A)) may also be important but have not found a way of using this yet.
I am trying to find the tangent space of SL(n,real) where A(0) is defined to be the identity matrix.
First of all I worked on the case when n=2 and found that the tangent space was
A = \left( \begin{array}{ccc}<br /> a & b \\<br /> c & -a<br /> \end{array} \right)
where a,b,c belong to the reals,
so I made the conjecture that for n in general, the tangent space would be the space of traceless matrices.
I attempted to prove this by showing that the tangent space and the space of traceless matrices were subsets of each other. Whilst I could show that an arbitary element of the tangent space is traceless, I could not show the converse.
Do I just need to try harder or is my conjecture just plain wrong?
PS. I used the standard result: d/dt (detA(0)) = tr(dA(0)/dt)
I have reason to believe that det(exp(A)) = exp(tr(A)) may also be important but have not found a way of using this yet.
