= Idempotent matrix
{title2=$P^2=P$}
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, $\left(\begin{smallmatrix}1&1\\0&0\end{smallmatrix}\right)$ 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.
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