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Vector Calculus

  1. Nov 9, 2005 #1

    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

    [tex]A = \left( \begin{array}{ccc}
    a & b \\
    c & -a
    \end{array} \right) [/tex]

    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.
  2. jcsd
  3. Nov 10, 2005 #2
    Done it

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