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).