I have 4 vectors $x_1$, $x_2$, $x_3$, and $x_4$. Let $x=[x_1^T, x_2^T, x_3^T, x_4^T]^T$. Then I have 6 matrices $A_{12}$, $A_{21}$, $A_{23}$, $A_{32}$, $A_{24}$, and $A_{42}$. Then, I define the block matrix \begin{align} A = \begin{pmatrix} A_{12} & -A_{21} & 0 & 0 \\ 0 & A_{23} & -A_{32} & 0 \\ 0 & A_{24} & 0 & -A_{42} \end{pmatrix} \end{align}
Let the set $E =$ {(1,2), (2,1), (2,3), (3,2), (2,4), (4,2)}, then I know that $$Q(A) = \frac{1}{2} \sum_{i=1}^4 \sum_{j \in E} ||A_{ij} x_i - A_{ji} x_j||^2 = x^T A^T A x.$$
I want to optimize $Q$ w.r.t the matrices $A_{ij}$ for $\forall (i,j) \in E$. If I run one step of gradient descent on the LHS, I can write: \begin{align} A_{ij}^{new} = A_{ij} - \alpha (A_{ij} x_i - A_{ji} x_j) x_i^T. \quad \quad \quad (1) \end{align}
Now, if I run step of gradient descent on the RHS, I get: \begin{align} A^{new} = A - 2 \alpha A x x^T. \quad \quad \quad (2) \end{align}
Unless I'm doing something wrong, I was expecting if I write the update (2) for each block, I should recover (1), but for some reason, the new matrix $A^{new}$ has a different block structure than $A$, giving rise to new blocks instead of the zero-blocks in $A$. Am I missing something or did I make a mistake along the way in my analysis?