**Readings Questions:**

You should read the section before
attempting any reading questions or problems. Remember that you are responsible
for all definitions and examples. **Also remember to keep the answers
to the reading questions on a separate piece of paper; these will be randomly
collected from time to time.**

1) Let
and .
Show that *q*(**x**)
is negative definite first without using Sylvester's theorem and then using
this theorem (theorem 2.5.2).

2) Consider .

a) Show that p(x,y) is positive semi-definite without
using Sylvester's theorem.

**Note: ** This shows that p(x,y) is ** not** positive
definite.

b) Show that

c) Why does this not contradict Sylvester's theorem?

**Problems:**

**page 148
1, 3, 4, 7, 10, 13, 15, 17, 20a, 21**