I have a error signal which I want to minimize using MSE. This error signal at time $k$ is a vector of length $3$: $e_k = C^{T} R_k - B^{T} A_k = [c_0, \ldots, c_{N_c-1}] \begin{bmatrix} r_{2k}\\ \vdots \\ r_{2k-N_c+1} \end{bmatrix} - [B_0, B_1, B_2]\begin{bmatrix} a_{k-d}\\ a_{k-d-1}\\ a_{k-d-2} \end{bmatrix} = [C^{T} - B^{T}] \begin{bmatrix} R_k\\ A_k \end{bmatrix} = W^{T} V_k$
where each $c_i, 0\leq i \leq N_c-1$ is $3 \times 3$, $r_{2k},\ldots, r_{2k-N_c+1}$ is $3 \times 1$, each $B_i, 0\leq i \leq 2$ is $3 \times 9$, and $a_{k-d}, a_{k-d-1}, a_{k-d-2}$ are $3 \times 1$ matrices.
Therefore,$\nabla_W \mathrm{MSE} = \nabla_W E(e_k^{T} e_k) = 2R_k W = 0$. Unfortunately this gives me $W=0$.
Any idea how else I can find my weight matrix $W$ which minimizes the error?
There is a certain defect in your setting. Of course if you model your error vector is a liner function of parameters $W$ the best solution will be $W=0$. The typical way to implement MSE is 1. Specify you measurement observable vector $y$. 2. Specify your model explaining $y$ as $W^T V$, where $V$ are some explanatory variables. 3. Minimize error $|y-W^TV|$.
Otherwise you cannot apply MSE.