Assume that matrices $\mathbf{A}$, $\mathbf{B}$, and $\mathbf{C}=\mathbf{A}\mathbf{B}$ are given. I aim to find matrices $\mathbf{E}_1$, $\mathbf{E}_2$, and $\mathbf{E}_3$ such that \begin{align} \operatorname{vec}(\mathbf{C}) \approx \mathbf{E}_3 \Big( \mathbf{E}_1 \operatorname{vec}(\mathbf{A}) \odot \mathbf{E}_2 \operatorname{vec}(\mathbf{B}) \Big), \end{align} where $\odot$ denotes the elementwise (Hadamard) product. My goal is to construct $\mathbf{E}_1$ and $\mathbf{E}_2$ with as few rows as possible, i.e., to minimize the intermediate dimension while maintaining a good approximation. However, I am unsure how to approach and formulate this problem in a way that makes it solvable. Do you have any suggestions?
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2$\begingroup$ Am I missing something or you might as well have started with vectors instead of vectorizing three matrices $\mathbf{A}, \mathbf{B}, \mathbf{C}$? $\endgroup$Claude Chaunier– Claude Chaunier2025-06-13 19:09:05 +00:00Commented Jun 13 at 19:09
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1 Answer
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If both $\mathbf{A}$ and $\mathbf{B}$ have any non-zero component, then you can easily take one-row matrices $E_1$ and $E_2$ that yield $E_1 \operatorname{vec}(\mathbf{A}) = E_2 \operatorname{vec}(\mathbf{B}) = (1) = E_1 \operatorname{vec}(\mathbf{A}) \odot E_2 \operatorname{vec}(\mathbf{B})$. Then simply take $E_3 = \operatorname{vec}(\mathbf{C})$. Am I missing anything ?
