Diagonal entries of $U$ in $LU$ factorisation of positive definite matrix

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Let $A\in M(n,\mathbb R)$ be a symmetric positive definite matrix. Let $L$ be a lower triangular matrix with real entries, all whose diagonal entries are $1$ and $LA$ is upper triangular.

Then, is it true that all the diagonal entries of $LA$ are positive ?

Definitely all the eigenvalues of $LA$ are non-zero, but I am not able to show that they are all positive.

Please help.

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1 Answer

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Let $LA = U$. By Sylvester's criterion a matrix is positive definite if and only if all of its leading principal minors (lpms) are positive. As $L$ is lower triangular, the lpms of $U$ are just the products of lpms of $L$ and $A$ (of the corresponding size). As all lpms of $L$ are equal to $1$, the lpms of $A$ and $U$ agree. Hence $U$ is positive definite.

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