Projection operator

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See also projection (linear algebra).

In mathematics, a projection operator P on a vector space is a linear transformation that is idempotent, that is, P² = P. Such transformations project any point in the vector space to a point in the subspace that is the image of the transformation.

Intuitively, a projection operator "picks out" entries in a vector, for example,

<math> p_{1,3,4} \begin{pmatrix} 3 \\ 97 \\ \pi \\ -17.73 \\ 10^{10^{10}} \end{pmatrix} = \begin{pmatrix} 3 \\ 0 \\ \pi \\ -17.73 \\ 0\end{pmatrix}. </math>

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 a matrix P which satisfies P² = P and P^* = P where P^* is the conjugate transpose of P (see projection (linear algebra)). The condition that P^* = P says P is a symmetric matrix if all of the entries in P are real.

In physics, the term projection operator usually means self-adjoint projection operator.

The only possible eigenvalues of a projection operator over a finite-dimensional real or complex vector space are 0 and 1.es:Operador de proyección