# Orthogonal 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 planes 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* and *n*×*k* is a matrix with more rows than columns, the *k* columns spanning a *k*-dimensional subspace of an *n*-dimensional space, then the matrix of the orthogonal projection onto the column space of *M* is

*M*is

**not a square matrix**but has more rows than columns!).

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

**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.

A related concept is used in technical drawing, where **orthogonal projection**, more correctly called orthographic projection, is drawing of the views of an object projected onto orthogonal planes. Commonly known views of this type are *plan* (*plan view*), *side view* and *elevation*.