SummaryWe deal with linear operators acting in a finite dimensional complex Hilbert space. We show that there exists a simple canonical form for projectors (not necessarily orthogonal) under unitary similarity. As a consequence we obtain a simple test for unitary similarity of projectors. IfP is a projector we show thatP andP* are unitarily similar. We also determine the isomorphism type of the algebra generated by the projectorsP andP*.