A projection matrix is an square matrix that gives a vector space projection from to a subspace . The columns of are the projections of the standard basis vectors, and is the image of . A square matrix is a projection matrix iff .
A projection matrix is orthogonal iff
(1)

where denotes the adjoint matrix of . A projection matrix is a symmetric matrix iff the vector space projection is orthogonal. In an orthogonal projection, any vector can be written , so
(2)

An example of a nonsymmetric projection matrix is
(3)

which projects onto the line .
The case of a complex vector space is analogous. A projection matrix is a Hermitian matrix iff the vector space projection satisfies
(4)

where the inner product is the Hermitian inner product. Projection operators play a role in quantum mechanics and quantum computing.
Any vector in is fixed by the projection matrix for any in . Consequently, a projection matrix has norm equal to one, unless ,
(5)

Let be a algebra. An element is called projection if and . For example, the real function defined by on and on is a projection in the algebra , where is assumed to be disconnected with two components and .