why's the matrix exponential of an anti self-adjoint matrix unitary? Let L be a matrix, (.,.) a unitary scalar product, t a real number.
Apparently, if L is anti self-adjoint (i.e. (Lv,w) = -(v,Lw)) then exp(tL) is unitary.
Could anyone please tell me why? Is this always true?

Anti self-adjoint for a matrix is the same as Skew-Hermitian, which means that $\overline{L^t}=-L$, and since $e^L e^{\overline{L^t}} = e^L e^{-L}=I$ you get the result you asked for. See http://en.wikipedia.org/wiki/Skew-hermitian  which also has an interpretation in terms of lie groups and algebras (Skew-Hermitian matrices form the lie algebra of U(n))
Edit : What I wrote before was confusing since $e^Ae^B = e^{A+B}$ is only true for matrices if $A$ and $B$ commute.
