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

Corners of Leavitt path algebras of finite graphs are Leavitt path algebras