## 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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