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. 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.

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.

The eigenvalues of a projection matrix must be 0 or 1.

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 $${\displaystyle P^{2}{\begin{bmatrix}x\\y\\z\end{bmatrix}}=P{\begin{bmatrix}x\\y\\0\end{bmatrix}}={\begin{bmatrix}x\\y\\0\end{bmatrix}}=P{\begin{bmatrix}x\\y\\z\end{bmatrix}}.}$$

Observing that ${\displaystyle P^{\mathrm {T} }=P}$ shows that the projection is an orthogonal projection.