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I am facing following equation:

$$ A * X + C \cdot X = D $$

with:

  • $A, C, D \in \mathbb{R}^{n \times n}$ some known matrices without any particular structure,
  • $X \in \mathbb{R}^{n \times n}$ the unknown matrix
  • "$\cdot$" the element-wise multiplication
  • "$*$" the cyclic convolution defined as: $A * X = \mathcal{F}^{-1}( \mathcal{F}(A) \cdot \mathcal{F}(X) ) $.

Is there a way to solve this? An analytic way would be fantastic but fast numerical methods are also welcome (for instance, I thought of minimizing the error... but not very elegant nor efficient).

I already tried to make some Fourier transforms to get rid of the convolution but unsuccessfully so far... Any help will be highly appreciated!!!

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  • $\begingroup$ you really want element-wise multiplication and not matrix multiplication? $\endgroup$
    – Carlo Beenakker
    Commented Dec 12, 2022 at 17:29
  • $\begingroup$ Correct, element-wise and not matrix multiplication! $\endgroup$
    – JannyBunny
    Commented Dec 12, 2022 at 17:31

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