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## Homework Statement

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

## Homework Equations

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.

Cheers,

W. =)