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I would like to construct a rectangular matrix which doesn't have a decomposition into a product of a sparse and a small matrices. It is easy to see that a random matrix doesn't have such a ...

Let $\mathbb{F}$ is a field. Consider an $n \times n$ matrix $\bf A$ over $\mathbb{F}$. $\bf A$ is called sparse matrix over $\mathbb{F}$ iff the number of non-zero entities of $\bf A$ be at most ...