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Does fixing diagonal entries and minimizing nuclear norm under weighted sum of entries conditions produce a rank $1$ matrix? I think the answer for this is no. At least could it be true in $2\times2$ ...

The nuclear norm minimization for the matrix completion problem is given by \begin{align} \textrm{minimize } \quad &\|X\|_{*}\\ \textrm{subject to } \quad & X_{ij}=M_{ij} \quad \forall (i,j)...