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If each row is a tensor product $v_i \otimes v_i$, then any vector $x$ that is a column-stacked version of a skew-symmetric matrix is in the kernel of $A$. For any $x$, you have $$(v_i \otimes v_i)x = v_i X v_i^T,$$ where $x = \mathrm{vec}(X)$ (see: https://en.wikipedia.org/wiki/Kronecker_product). If $X = -X^T$ then $$v_i X v_i^T = -v_i X^T v_i^T = -v_i X ...


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