Write linear constraints into linear transformation

Question

Suppose we have the following block matrix $$ X=\begin{bmatrix} X_{11} & X_{12} & X_{13}\\ X_{21} & X_{22} & X_{23}\\ X_{31} &X_{32} & X_{33} \end{bmatrix} $$ where $X_{ij} \in \mathbb{R}^{n \times n},i,j=1,2,3$.For this block matrix,we have following constraints $$ X_{12}+X_{21}=0\\ X_{13}+X_{13}=0\\ X_{23}+X_{32}=0 $$ Can we write the constraints as $\mathcal{A}(X)=0$,where $\mathcal{A}$ is a linear operator

Answer 1

Define $\mathcal{A}: \mathbb R^{3n \times3n} \to \mathbb R^{3n \times n}$ by

$\mathcal{A}(X)= \begin{bmatrix} X_{12} + X_{21} \\ X_{13} + X_{31} \\ X_{23} + X_{32} \end{bmatrix}$