Uniqueness of Cholesky Decomposition

Question

Suppose A=LL^T, where L is a lower triangular matrix whose diagonal entries are all positive. If another lower triangular matrix P also satisfies A=PP^T, and the diagonal entries of P are also positive, show that P=L.

This are my steps.

LL^T = PP^T
L=PP^TL^-T
P^-1L=(L^-1P)^T=D where D is a diagonal matrix.
L=DP

But I am stuck after this. I have not used the property that the diagonal entries of L and P are positive. Does this imply that D=I?

Your help is greatly appreciated.

Answer 1

$$P^{-1}L=D$$

$$L=PD$$

Hence substitute that into $LL^T=PP^T$,

$PDD^TP^T=PP^T$, we have $D^2=I$.

Now use the positivity of the diagonal entry to conclude that $D=I$.