Idempotent matrix

Definition

Let $R$ be a ring and let $P$ be an $n \times n$ square matrix over $R$. The matrix $P$ is idempotent when $P^2 = P$.

Remarks

The matrices $0$, $I_n$ and $\begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix}$ are idempotent. Over a field, an idempotent matrix is the matrix of a projection onto its image along its kernel. If $P$ is idempotent, then so is $I - P$. Over a field, the eigenvalues of an idempotent matrix are in $\{0, 1\}$, and its rank equals its trace in characteristic $0$.

Explanations

L1 human

An idempotent matrix does nothing new when we apply it twice. A projection onto a line is the main example: once a point is on the line, projecting again keeps it there. So $P^2 = P$.