When factorizing a real symmetric matrix $A$ into $LDL^T$, the matrix $D$ can have 1x1 or 2x2 blocks on the diagonal. A condition for $A$ to be positive-definite is that all 1x1 and 2x2 blocks of $D$ are positive-definite.

As in this question, I would like to know it would be possible to project $A$ into its positive-eigenvalue subspace from its $LDL^T$ factorization.

For example, I am wondering if defining $D'$ by replacing all non-positive blocks of $D$ by 0 would result in this projection for $A' = LD'L^T$.



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