Does a closed form solution exist for $\beta$? I am having trouble trying to solve this equation for an unknown matrix $\beta$.$$K = \sum_{i=1}^n \tau_i \sum_{j=1}^p (A_j + A_j^T)\beta(C_j + C_j^T)$$ where $\tau_i$ is a scalar constant, and $A_j$, $C_j$, K are matrices of constants.
 A: I don't believe that this can be solved using only typical matrix operations (e.g. additions multiplications and transposes). However, it can be solved using vectorization.
Notation: Let $\operatorname{vec}(A)$ denote the (column-major) vectorization of $A$. Let $\otimes$ denote the Kronecker product. Denote $B_j = A_j + A_j^T$, $D_j = C_j + C_j^T$; note that these matrices are symmetric.
We can vectorize both sides of the equation and simplify as follows.
$$
K = \sum_i \tau_i \sum_j B_j \beta D_j\\
\operatorname{vec}(K) = \operatorname{vec}\left[\sum_i \tau_i \sum_j B_j \beta D_j \right]\\
\operatorname{vec}(K) = \sum_i \tau_i \sum_j \operatorname{vec}(B_j \beta D_j)\\
\operatorname{vec}(K) = \sum_i \tau_i \sum_j (D_j \otimes B_j)\operatorname{vec}(\beta).
$$
We can solve this for $\operatorname{vec}(\beta)$ to find that
$$
\operatorname{vec}(\beta) = \left[\sum_i \tau_i \sum_j (D_j \otimes B_j)\right]^{-1}\operatorname{vec}(K) = 
\frac{\left[\sum_j (D_j \otimes B_j)\right]^{-1}\operatorname{vec}(K)}{\sum_i \tau_i }.
$$
Obtain $\beta$ by "unvectorizing" $\operatorname{vec}(\beta)$.
