Let |||| be any norm on Rn. (a) Show that q(x) is a positive definite quadratic form

Question:

Let ||∙|| be any norm on Rn.
(a) Show that q(x) is a positive definite quadratic form if and only if q(u) > 0 for all unit vectors, ||u|| = 1.
(b) Prove that if S = ST is any symmetric matrix, then K = S + c I > 0 is positive definite for c»0 sufficiently large.