Projection operator - Definition and Overview

In mathematics, a projection operator on a vector space is an idempotent linear transformation. Such transformations project any point in the vector space to a point in the subspace that is the image of the transformation. In an inner product space, such an operator is an orthogonal projection if and only if it is self-adjoint. In finite-dimensional inner product spaces, an orthogonal projection matrix is one whose matrix M satisfies M² = M and M ′ = M where M ′ is the conjugate transpose of M (see projection (linear algebra)). The condition that M ′ = M says M is a symmetric matrix if all of the entries in M are real. In physics, the term projection operator usually means self-adjoint projection operator.