I believe I understand what the exponential map does, in informal terms. However I cannot relate this to the equation I see before me in the papers.
The exponential mapping function for a symmetric matrix $A$ belonging to the tangent space at $X$, $T_x\mathcal{P}_n$ is: $S_A = Exp(A) = X^\frac{1}{2} exp(X^{-\frac{1}{2}} A X^{-\frac{1}{2}}) X^\frac{1}{2} $
where $Exp()$ is the exponential map and $exp()$ is the matrix exponential.
My informal understanding is that the exponential map takes a tangent vector and travels along the geodesic whose tangent is that vector. The point that you end up at $S_A$ is the same distance from along the geodesic from X as the length of the tangent vector.
Presuming that my intuition is correct - how do I relate this to the equation? Why is this $X^\frac{1}{2}$ decomposition going on..? What purpose does the exponential of the matrix serve?