  We achieve an extremely useful description (up to isomorphism) of the Leavitt
path algebra $L_K(E)$ of a finite graph $E$ with coefficients in a field $K$ as
a direct sum of matrix rings over $K$, direct sum with a corner of the Leavitt
path algebra $L_K(F)$ of a graph $F$ for which every regular vertex is the base
of a loop. Moreover, in this case one may transform the graph $E$ into the
graph $F$ via some step-by-step procedure, using the "source elimination" and
"collapsing" processes. We use this to establish the main result of the
article, that every nonzero corner of a Leavitt path algebra of a finite graph
is isomorphic to a Leavitt path algebra. Indeed, we prove a more general
result, to wit, that the endomorphism ring of any nonzero finitely generated
projective $L_K(E)$-module is isomorphic to the Leavitt path algebra of a graph
explicitly constructed from $E$. Consequently, this yields in particular that
every unital $K$-algebra which is Morita equivalent to a Leavitt path algebra
is indeed isomorphic to a Leavitt path algebra.
