Idempotent matrix

In linear algebra, an idempotent matrix is a matrix which, when multiplied by itself, yields itself.^[1][2] That is, the matrix ${\displaystyle A}$ is idempotent if and only if ${\displaystyle A^{2}=A}$. For this product ${\displaystyle A^{2}}$ to be defined, ${\displaystyle A}$ must necessarily be a square matrix. Viewed this way, idempotent matrices are idempotent elements of matrix rings.