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I have encountered the equation $\mathbf{I} = \sum_{k=1}^{m} \mathbf{X} \mathbf{B}_k \mathbf{X}^T\mathbf{A}_k$, recently.

All matrices are of dimension $n \times n$.

Is it assigned a special name?

I'm trying to find a closed form or efficient approximate solution for it.

Do you know any solution or reference?

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    $\begingroup$ in relation to your previous similar question: please don't delete questions after you have received a lot of feedback, in consideration for others who might benefit from that feedback --- mathoverflow.net/questions/255081/… $\endgroup$
    – Carlo Beenakker
    Commented Nov 20, 2016 at 12:20
  • $\begingroup$ Not sure if it helps but with a formula from en.wikipedia.org/wiki/Vectorization_(mathematics) the equation becomes equivalent to $$vec(X^{-1})=\left(\sum_k A_k^T \otimes B_k\right)vec(X^T)$$ where $vec\colon \mathbb{R}^{3\times3}\rightarrow \mathbb{R}^{9}$ denotes the vectorization. $\endgroup$
    – Markus Sprecher
    Commented Nov 20, 2016 at 18:13

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