Idempotent Matrix

Definition

$A^2=A$ A matrix whose squared matrix is itself.

Facts

Eigenvalues of idempotent matrix is either $0$ or $1$.

Let $A$ be a symmetric matrix. Then, $A$ has $r$ eigenvalues equal to $1$ and the rest zero $\iff A^2=A,\mathrm{rank}(A)=r$.

Let $A$ be an idempotent matrix, then $\mathrm{rank}(A)=\mathrm{tr}(A)$

If $A$ is idempotent, then $I-A$ also idempotent.