Orthogonal projection

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This article is about the use of "orthogonal projection" in linear algebra. For the use of orthogonal projections in technical drawing, see orthographic projection.

In geometry, an orthogonal projection of a k-dimensional object onto a d-dimensional hyperplane (d < k) is obtained by intersections of (k − d)- dimensional hyperplanes drawn through the points of an object orthogonally to the d-hyperplane. In particular, an orthogonal projection of a three-dimensional object onto a plane is obtained by intersections of lines drawn through all points of the object orthogonally to the plane of projection.

If such a projection leaves the origin fixed, it is a self-adjoint idempotent linear transformation; its matrix is a symmetric idempotent matrix. Conversely, every symmetric idempotent matrix is the matrix of the orthogonal projection onto its own column space. If M is a k×d matrix with d < k, the d columns spanning a d-dimensional subspace, then the matrix of the orthogonal projection onto the column space of M is

(and before leaping to the conclusion that this must be an identity matrix, remember that M is not a square matrix but has more rows than columns!).

If the basis is orthonormal, the projection can be simplified to

In functional analysis, the geometric notion is generalized as follows. An orthogonal projection is a bounded operator on a Hilbert space H which is self-adjoint and idempotent. It maps each vector v in H to the closest point of PH to v. PH is the range of P and it is a closed subspace of H.

See also spectral theorem, orthogonal matrix, and projection (linear algebra).