Projection Matrix
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],](/National_Library/im_/http://mathworld.wolfram.com/images/equations/ProjectionMatrix/NumberedEquation3.gif) |
(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 
.
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