Proving the Existence of Cholesky Decomposition: Lemma on Positive Matrices

[Doom of Doom]

I'm supposed to prove this step as part of my proof for existence of Cholesky Decomposition. I can see how to use it in my proof, but I can't seem to be able to prove this lemma:

For any positive (nxn) matrix A and any non-singular (nxn) matrix X, prove that

B=X^{\dagger}A X

is positive.

____

Let X=\left(x_{1}, x_{2}, \ldots, x_{n}\right), where all xi are n-vectors.

I see that
b_{i,j}=x_{i}^{\dagger}Ax_{j},
and thus all of the diagonal elements of B are positive (from the definition of a positive matrix).

But where do I go from there?

 

[yyat]

I think you mean http://en.wikipedia.org/wiki/Positive-definite_matrix" .

In that case, you cannot conclude that the diagonal elements of B are positive.
Try to show instead directly that v^TBv>0 for v\neq 0 (this is a one-liner), then B is positive definite by definition.

 

[Doom of Doom]

Dur! Yeah, i meant positive definite.

I see it now. It's so easy...

For arbitrary vector v, let z = Xv, and thus z^{\dagger}=v^{\dagger}X^{\dagger}.

So v^{\dagger}Bv=v^{\dagger}X^{\dagger}AXv=z^{\dagger}Az>0.