How would I go about solving the matrix equation $\sum_iA_i^TA_iXB_iB_i^T=\sum_iA_i^TC_iB_i^T$ for $X$?
The simplest thing to do would be to, of course, consider $Y=\sum_iA_i^TC_iB_i^T$, vectorise and use Krönecker products. \begin{align*} vec(\sum_iA_i^TA_iXB_iB_i^T) &= vec(Y)\\ \implies \sum_ivec(A_i^TA_iXB_iB_i^T) &= vec(Y)\\ \implies \sum_i\big((B_iB_i^T)\otimes(A_i^TA_i)\big)vec(X) &= vec(Y)\\ \implies vec(X) &= \Big(\sum_i\big((B_iB_i^T)\otimes(A_i^TA_i)\big)\Big)^{-1}vec(Y)\\ \end{align*}
I want to solve it without inverting, or if possible even computing any Krönecker products. Is there a computationally efficient way to solve it? The major obstacle seems to be the summation, as if it were not there, I could simply distribute the inverse. Is there a way to write the sum of Krönecker products as the Krönecker product of sums?
