Geometry and algebra of projectors
Projectors are geometrically so simple that they should have been discussed somewhere in the beginning of this course. I am giving them now because the applications are more advanced.
Motivating example
Let be the
-axis and
the
-axis on the plane. Let
be the projector onto
along
and let
be the projector onto
along
This geometry translates into the following definitions:
The theory is modeled on the following observations.
a) leaves the elements of
unchanged and sends to zero all elements of
b) is the image of
and
is the null space of
c) Any element of the image of is orthogonal to any element of the image of
d) Any can be represented as
It follows that
For more simple examples, see my post on conditional expectations.
Formal approach
Definition 1. A square matrix is called a projector if it satisfies two conditions: 1)
(
is idempotent; for some reason, students remember this term better than others) and 2)
(
is symmetric).
Exercise 1. Denote the set of points
that are left unchanged by
Then
is the image of
(and therefore a subspace).
Proof. Indeed, the image of consists of points
For any such
we have
so
belongs to
Conversely, any element of
is seen to belong to the image of
Exercise 2. a) The null space and image of are orthogonal. b) We have an orthogonal decomposition
Proof. a) If and
then
and by Exercise 1
Therefore
This shows that
b) For any write
Here
and
because
Exercise 3. a) Along with the matrix
is also a projector. b)
and
Proof. a) is idempotent:
b)
is symmetric:
b) By Exercise 2
Since this equation implies
It follows that, as with the set
is the image of
and it consists of points that are not changed by
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