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.
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.
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