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
![P=[0 1; 0 1],](http://mathworld.wolfram.com/images/equations/ProjectionMatrix/NumberedEquation3.svg) | (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 
.
