  The set of morphisms mapping any Sturmian sequence to a Sturmian sequence
forms together with composition the so-called monoid of Sturm. For this monoid,
we defne a faithful representation by $(3\times 3)$-matrices with integer
entries. We find three convex cones in $\mathbb{R}^3$ and show that a matrix $R
\in Sl(\mathbb{Z},3)$ is a matrix representing a Sturmian morphism if the three
cones are invariant under multiplication by $R$ or $R^{-1}$. This property
offers a new tool to study Sturmian sequences. We provide alternative proofs of
four known results on Sturmian sequences fixed by a primitive morphism and a
new result concerning the square root of a Sturmian sequence.
