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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.
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