Linear Algebra: Symmetric/Positive Definite problem

[Scootertaj]

1. Let A$\in$R^nxn be a symmetric matrix, and assume that there exists a matrix B$\in$R^mxn such that A=B^TB.
a) Show that A is positive semidefinite
B) Show that if B has full rank, then A is positive definite
2. Homework Equations :
This is for an operations research class, so most of the definitions revolve around minimizing/maximizing.
However, alternate definitions:
Positive Definite: A is positive definite if for all non-null vectors h, h^TAh > 0.
Symmetric: if A^T=A.
Semidefinite: h^TAh ≥ 0

The Attempt at a Solution

Here's some work:
A^T = A ; A = B^TB.
So, A^T = B^TB → A^TA = B^TBA = AA = A².
So, A^TA ≥ 0.
But, that's not quite what I want.

 

[Dick]

$h^T A h=h^T B^T B h$. That's the inner product $(B h)^T (B h)$. Use the properties of the inner product.

 

[Scootertaj]

Dick,
I don't recall any specific properties of inner products that would help except <x,x> >= 0. But, I don't see how that applies.

 

[Dick]

Dick,
I don't recall any specific properties of inner products that would help except <x,x> >= 0. But, I don't see how that applies.

And <x,x>=0 only if x=0. I think it applies a lot. $(B h)^T (B h)=<Bh, Bh>$.

 

[Scootertaj]

D'oh! I must be too tired, completely forgot that the inner product would be the same as doing the transpose first.
Thank you a lot Dick, you always seem to help out a lot.