Editing Projection (linear algebra) (section)

== Applications and further considerations ==

Projections (orthogonal and otherwise) play a major role in [[algorithm]]s for certain linear algebra problems:
* [[QR decomposition]] (see [[Householder transformation]] and [[Gram–Schmidt decomposition]]);
* [[Singular value decomposition]]
* Reduction to [[Hessenberg matrix|Hessenberg]] form (the first step in many [[eigenvalue algorithm]]s)
* [[Linear regression]]
* Projective elements of matrix algebras are used in the construction of certain K-groups in [[Operator K-theory]]

As stated above, projections are a special case of idempotents. Analytically, orthogonal projections are non-commutative generalizations of [[characteristic polynomial|characteristic functions]]. Idempotents are used in classifying, for instance, [[semisimple algebra]]s, while [[measure theory]] begins with considering characteristic functions of [[measurable set]]s. Therefore, as one can imagine, projections are very often encountered in the context of [[operator algebra]]s. In particular, a [[von Neumann algebra]] is generated by its complete [[lattice (order)|lattice]] of projections.