In the literature, several canonical decompositions of a system $\{C,A,B\}$ have been derived for different applications. In this paper we propose a canonical decomposition of a right invertible system $\{C,A,B\}$. From this decomposition, we study the Smith form of the matrix pencil $P(s)$ to find out the finite zeros and infinite zeros of $P(s)$, the range of the ranks of $P(s)$ for $s\in\mathbb{C}$, and the controllability and the invariant quantities of the right invertible system. In a separate paper, we apply this canonical decomposition of the right invertible system $\{C,A,B\}$ to study the decoupling and pole assignment problems.