A counterexample to Huang's weak majorization conjecture

We give a counterexample to a weak majorization problem, proposed by Z. Huang (Linear Algebra Appl., 434 (2) (2011) 457–462), for singular values of Hadamard products of nonnegative matrices. The problem asks whether, for all nonnegative matrices $A$ and $B$, \begin{equation*} \bigl{s_j^2(A\Had B)\bigr} \prec_w \bigl{s_j(A\Had A)s_j(B\Had B)\bigr} \end{equation*} holds, where $\text{Had}$ denotes the Hadamard product, ${\prec }_w$ means weak majorization, and $s_j(\cdot )$ is the $j$th largest singular value of a matrix. To answer this problem, we construct two $3\times 3$ entrywise positive symmetric matrices. We give rigorous upper and lower bounds for the relevant singular-value sums by Sturm’s theorem, and show that the inequality fails for the partial sum with $k=2$.