  Inspired by an interesting idea of Cai and Zhang, we formulate and prove the
convex $k$-sparse decomposition of vectors which is invariant with respect to
$\ell_1$ norm. This result fits well in discussing compressed sensing problems
under RIP, but we believe it also has independent interest. As an application,
a simple derivation of the RIP recovery condition $\delta_k+\theta_{k,k} < 1$
is presented.
