Finding the Tangent Space of SL(n,real) with A(0) being the Identity Matrix

[Diophantus]

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

$$A = \left( \begin{array}{ccc} a & b \\ c & -a \end{array} \right)$$

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.

 

[Diophantus]

Done it

Not to worry, I have solved it by myself. It must have just been too hard for you Americans.