B is nonsingular -- prove B(transpose)B is positive definite.

[Adgorn]

Homework Statement

Suppose B is a real nonsingular matrix. Show that: (a) B^tB is symmetric and (b) B^tB is positive definite

2. Homework Equations
N/A

The Attempt at a Solution

I have managed to prove (a) by showing that elements that are symmetric on the diagonal are equal. However I have no idea how to prove B. I've tried to express [c_ij] with sigma notation with no sucess. I've also tried to apply the rules of matrices to try and show that u^tB^tBu is bigger than zero with no sucess as well (perhaps (Bu)^t=uB?...). Any help would be greatly appriciated.

 

[pasmith]

Well, $u^TB^TBu = (Bu)^T(Bu)$ and you haven't yet needed the condition that $B$ is nonsingular...

 

[Adgorn]

Well, $u^TB^TBu = (Bu)^T(Bu)$ and you haven't yet needed the condition that $B$ is nonsingular...

Oh wow that rule completely slipped out of my mind. Thank you for your help.