a projection operator P is a matrix such that P2 = P. One property of the matrix P is that the matrix (I -P) is also a projection operator.

The reason why these matrices are called projection operators are because every vector in Rn multiplied by P is projected into a subspace of R. In fact, Px is always in a subspace S that is orthogonal (some use "perpendicular") to a subspace L that any vector will be projected to if multiplied from (I-P) (ie (I-P)x will always be in the subspace L, which is orthogonal to S).

Projectors are important concepts in iterative methods for solving systems of linear equations.