In linear algebra and functional analysis, a projection is a linear transformation P {\displaystyle P} from a vector space to itself (an endomorphism) such that P ∘ P = P {\displaystyle P\circ P=P} . That is, whenever P {\displaystyle P} is applied twice to any vector, it gives the same result as if it were applied once (i.e. P {\displaystyle P} is idempotent). It leaves its image unchanged. This definition of "projection" formalizes and generalizes the idea of graphical projection. One can also consider the effect of a projection on a geometrical object by examining the effect of the projection on points in the object.
Definitions A projection on a vector space V {\displaystyle V} is a linear operator P : V → V {\displaystyle P\colon V\to V} such that P 2 = P {\displaystyle P^{2}=P} . When V {\displaystyle V} has an inner product and is complete, i.e. when V {\displaystyle V} is a Hilbert space, the concept of orthogonality can be used. A projection P {\displaystyle P} on a Hilbert space V {\displaystyle V} is called an orthogonal projection if it satisfies ⟨ P x , y ⟩ = ⟨ x , P y ⟩ {\displaystyle \langle P\mathbf {x} ,\mathbf {y} \rangle =\langle \mathbf {x} ,P\mathbf {y} \rangle } for all x , y ∈ V {\displaystyle \mathbf {x} ,\mathbf {y} \in V} . A projection on a Hilbert space that is not orthogonal is called an oblique projection.
Projection matrix A square matrix P {\displaystyle P} is called a projection matrix if it is equal to its square, i.e. if P 2 = P {\displaystyle P^{2}=P} . A square matrix P {\displaystyle P} is called an orthogonal projection matrix if P 2 = P = P T {\displaystyle P^{2}=P=P^{\mathrm {T} }} for a real matrix, and respectively P 2 = P = P ∗ {\displaystyle P^{2}=P=P^{*}} for a complex matrix, where P T {\displaystyle P^{\mathrm {T} }} denotes the transpose of P {\displaystyle P} and P ∗ {\displaystyle P^{*}} denotes the adjoint or Hermitian transpose of P {\displaystyle P} . A projection matrix that is not an orthogonal projection matrix is called an oblique projection matrix. The eigenvalues of a projection matrix must be 0 or 1.
Examples
Orthogonal projection For example, the function which maps the point ( x , y , z ) {\displaystyle (x,y,z)} in three-dimensional space R 3 {\displaystyle \mathbb {R} ^{3}} to the point ( x , y , 0 ) {\displaystyle (x,y,0)} is an orthogonal projection onto the xy-plane. This function is represented by the matrix
P = [ 1 0 0 0 1 0 0 0 0 ] . {\displaystyle P={\begin{bmatrix}1&0&0\\0&1&0\\0&0&0\end{bmatrix}}.}
The action of this matrix on an arbitrary vector is
P [ x y z ] = [ x y 0 ] . {\displaystyle P{\begin{bmatrix}x\\y\\z\end{bmatrix}}={\begin{bmatrix}x\\y\\0\end{bmatrix}}.}
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