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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?

Your help is greatly appreciated.

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  • $\begingroup$ 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. $\endgroup$ – xbh Sep 17 '18 at 1:45
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$$P^{-1}L=D$$

$$L=PD$$

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 $D=I$.

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  • $\begingroup$ Oh thanks for pointing out my mistake. $\endgroup$ – taupi Sep 20 '18 at 6:28

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