Given a real symmetric matrix A, prove that:
a) A is positive definite if and only if A = (B^T)B for some real invertible matrix B
b) A is positive semidefinite if and only if there exists a (possibly singular) real matrix Q such that A = (Q^T)Q
quadratic form q(x) = a1*x1^2 + ... + an*xn^2
And possibly the principal axis theorem.
The Attempt at a Solution
for part a) I think I know how to show that A is pos def if you assume A = (B^T)B:
(P^T)AP = D = (P^T)(B^T)BP = ((PB)^T)PB which implies the diagonal entries of the diagonal matrix D are positive since row of (A^T) = column of A.
I'm not sure how to do it the other way, however. Any hints or advice would be appreciated.