A square matrix, $\mathbf{A}$, is idempotent if it is equal to its square, i.e., $\mathbf{A}^2=\mathbf{A}$ [Ber18, Definition 4.1.1].

Notable properties:

• Every idempotent matrix is necessarily a projector [Mey00, (5.9.13)].
• $\mathbf{A}$ is idempotent $\implies \text{rank}(\mathbf{A})=\text{tr}(\mathbf{A})$ [Ber18, Proposition 8.1.7].
• $\mathbf{A}$ is idempotent $\iff$ eigenvalues of $\mathbf{A}$ are either 0 or 1 [Loc08].