# Uniqueness of Cholesky Decomposition

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

This are my steps.

LLT = PPT
L=PPTL-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?

• Hello there. We encourage users to type math expressions using MathJax, so it would be better if you take some time to read this tutorial. – xbh Sep 17 '18 at 1:45

$$P^{-1}L=D$$
$$L=PD$$
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 $D=I$.