Idempotent function ($f\circ f=f$)

A self-map $f:X\to X$ is idempotent if its function composition with itself leaves it unchanged: $f(f(x))=f(x)$ for every $x$. Equivalently, every element of its image of a function is a fixed point. For example, the function on $\{a,b,c\}$ sending $a$ to $b$, $b$ to $b$, and $c$ to $c$ is idempotent. A second application makes no further change.

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.