I quote the a textbook, which says the following:
It is easily checked that the exponential of a Hamiltonian matrix
$$ g=exp(\phi\cdot\mathbf{T}) $$ is a symplectic matrix; Lie group elements are related to the Lie algebra elements by exponentiation.
I looked for a more rigorous version of this for more clarity, but after spending a few hours, could not find what I'm looking for. I did find a Wikipedia entry, which was even more vague that this.
More concretely, my question is this: quite obviously, in the above quote, $g\in Sp(2n)$ and $\mathbf{T}\in sp(2n)$ for some number of canonical variables $n$, but is it the case that parameter $\phi$ is simply some $\phi\in\mathbb{R}$? Or is it something more complicated, like a tensor?
