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],](/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 
.
