Chapter 8.2, Problem 53E

Consider a quadratic form $q\left(x_1,\mathrm{...},x_n\right)$ with symmetric matrix A. For two integers i and j with $1\le i<j\le n$ , we define the function $p\left(x,y\right)=q\left(0,\mathrm{...0},\underset{i\text{th}}{\underset{︸}{x}},0,\mathrm{...},0,\underset{j\text{th}}{\underset{︸}{y}},0,\mathrm{...},0\right)$ .
a. Show that p is a quadratic form, with matrix $\left[\begin{array}{cc}{a}_{ii}& a_{ij}\\ a_{ji}& a_{jj}\end{array}\right]$ .
b. If q is positive definite, show that p is positive definite as well.
c. If q is positive semidefinite, show that p is positive semidefinite as well.
d. Give an example where q is indefinite, but p is positive definite.