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 transpose 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 nonzero
projection matrix 
has Euclidean operator norm at least one,
 |
(5)
|
Equality holds for a nonzero orthogonal projection. The nonsymmetric example above instead has operator
norm 
.
Let 
be a C*-algebra. An element 
is called a projection if 
and 
. For example, the real-valued
function 
defined by 
on 
and 
on 
is a projection in the C*-algebra 
of continuous
functions, where 
is a compact T2-space
and is disconnected with two components 
and 
.
