Editing Projection (linear algebra) (section)

==Definitions==
A '''projection''' on a vector space <math>V</math> is a linear operator <math>P\colon V \to V</math> such that <math>P^2 = P</math>.

When <math>V</math> has an [[inner product]] and is [[complete metric space|complete]], i.e. when <math>V</math> is a [[Hilbert space]], the concept of [[orthogonality]] can be used. A projection <math>P</math> on a Hilbert space <math>V</math> is called an '''orthogonal projection''' if it satisfies <math>\langle P \mathbf x, \mathbf y \rangle = \langle \mathbf x, P \mathbf y \rangle</math> for all <math>\mathbf x, \mathbf y \in V</math>. A projection on a Hilbert space that is not orthogonal is called an '''oblique projection'''.

===Projection matrix===
* A [[square matrix]] <math>P</math> is called a '''projection matrix''' if it is equal to its square, i.e. if <math>P^2 = P</math>.<ref name="HornJohnson">{{cite book |title=Matrix Analysis, second edition |first1=Roger A. |last1=Horn |first2=Charles R. |last2=Johnson |isbn=9780521839402 |publisher=Cambridge University Press|year=2013}}</ref>{{rp|p. 38}}
* A square matrix <math>P</math> is called an '''orthogonal projection matrix''' if <math>P^2 = P = P^{\mathrm T}</math> for a [[real number|real]] [[matrix (mathematics)|matrix]], and respectively <math>P^2 = P = P^{*}</math> for a [[complex number|complex]] matrix, where <math>P^{\mathrm T}</math> denotes the [[transpose]] of <math>P</math> and <math>P^{*}</math> denotes the adjoint or [[Hermitian transpose]]  of <math>P</math>.<ref name="HornJohnson">{{cite book |title=Matrix Analysis, second edition |first1=Roger A. |last1=Horn |first2=Charles R. |last2=Johnson |isbn=9780521839402 |publisher=Cambridge University Press|year=2013}}</ref>{{rp|p. 223}}
* A projection matrix that is not an orthogonal projection matrix is called an '''oblique projection matrix'''.
The [[eigenvalue]]s of a projection matrix must be 0 or 1.