Idempotent matrix ($P^2=P$)

An idempotent matrix is a square matrix $P$ with $P^2=P$. Its linear map fixes every vector in its image and sends its kernel to zero. Every vector splits uniquely as $v=Pv+(v-Pv)$, giving a direct sum of the image and the kernel.

Consequently $P$ is diagonalizable over any field, with only the eigenvalues zero and one. For example, $\left(\begin{array}{cc}1& 1\\ 0& 0\end{array}\right)$ is idempotent but not symmetric. An idempotent matrix is an orthogonal projection in a real Euclidean inner product precisely when it is also symmetric; idempotence alone gives an oblique projection.