If I have \begin{align} B = {\arg \min}_{A\in\text{Sym}^d} \sum_j (\text{tr}(A X_j) -f_j)^2 \end{align} where $f_j>0$ and $X_j$ are positive semi-definite and there are sufficient $j$ that this is not an interpolation, and $\text{Sym}^d$ is the space of symmetric $d\times d$ matrices. Can I show that $B$ must be negative definite?
2026-03-25 15:54:44.1774454084
Least squares for trace of matrix product
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