# Homework Help: Linear Algebra: Symmetric/Positive Definite problem

1. Jan 29, 2012

### Scootertaj

1. Let A$\in$Rnxn be a symmetric matrix, and assume that there exists a matrix B$\in$Rmxn such that A=BTB.
a) Show that A is positive semidefinite
B) Show that if B has full rank, then A is positive definite

2. Relevant 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, hTAh > 0.
Symmetric: if AT=A.
Semidefinite: hTAh ≥ 0

3. The attempt at a solution
Here's some work:
AT = A ; A = BTB.
So, AT = BTB → ATA = BTBA = AA = A2.
So, ATA ≥ 0.
But, that's not quite what I want.

Last edited: Jan 29, 2012
2. Jan 29, 2012

### 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.

Last edited: Jan 29, 2012
3. Jan 29, 2012

### 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.

4. Jan 29, 2012

### Dick

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

5. Jan 30, 2012

### 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.