# Learning From Data – A Short Course: Exercise 3.3

Page 87, Exercise 3.3:

Consider the hat matrix , where is an by matrix, and is invertible.

(a) Show that is symmetric.

(b) Show that for any positive integer .

(c) If is the identity matrix of size , show that for any positive integer .

(d) Show that , where the trace is the sum of diagonal elements.

**(a) and (d)** Forgot the linear algebra, will come back to it later (or never, lol).

**(b)** Base cases:

- :
- : (stated in book, verify later)

Induction step for :

Due to :

We also have , so:

**(c)** We observe that:

The rest of the proof is similar to (b).