In linear algebra and functional analysis, a projection is a linear transformation ${\displaystyle P}$ from a vector space to itself (an endomorphism) such that ${\displaystyle P\circ P=P}$. That is, whenever ${\displaystyle P}$ is applied twice to any vector, it gives the same result as if it were applied once (i.e. ${\displaystyle P}$ is idempotent). It leaves its image unchanged.^[1] 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 ${\displaystyle V}$ is a linear operator ${\displaystyle P\colon V\to V}$ such that ${\displaystyle P^{2}=P}$.

When ${\displaystyle V}$ has an inner product and is complete, i.e. when ${\displaystyle V}$ is a Hilbert space, the concept of orthogonality can be used. A projection ${\displaystyle P}$ on a Hilbert space ${\displaystyle V}$ is called an orthogonal projection if it satisfies ${\displaystyle \langle P\mathbf {x} ,\mathbf {y} \rangle =\langle \mathbf {x} ,P\mathbf {y} \rangle }$ for all ${\displaystyle \mathbf {x} ,\mathbf {y} \in V}$. A projection on a Hilbert space that is not orthogonal is called an oblique projection.

Projection matrix

• A square matrix ${\displaystyle P}$ is called a projection matrix if it is equal to its square, i.e. if ${\displaystyle P^{2}=P}$.^{[2]: p. 38}
• A square matrix ${\displaystyle P}$ is called an orthogonal projection matrix if ${\displaystyle P^{2}=P=P^{\mathrm {T} }}$ for a real matrix, and respectively ${\displaystyle P^{2}=P=P^{*}}$ for a complex matrix, where ${\displaystyle P^{\mathrm {T} }}$ denotes the transpose of ${\displaystyle P}$ and ${\displaystyle P^{*}}$ denotes the adjoint or Hermitian transpose of ${\displaystyle P}$.^{[2]: p. 223}
• 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 ${\displaystyle (x,y,z)}$ in three-dimensional space ${\displaystyle \mathbb {R} ^{3}}$ to the point ${\displaystyle (x,y,0)}$ is an orthogonal projection onto the xy-plane. This function is represented by the matrix

$${\displaystyle P={\begin{bmatrix}1&0&0\\0&1&0\\0&0&0\end{bmatrix}}.}$$

The action of this matrix on an arbitrary vector is

$${\displaystyle P{\begin{bmatrix}x\\y\\z\end{bmatrix}}={\begin{bmatrix}x\\y\\0\end{bmatrix}}.}$$

To see that ${\displaystyle P}$ is indeed a projection, i.e., ${\displaystyle P=P^{2}}$, we compute