We can generalize the above equations to $$x_i^{(k+1)}=(b_i-\sum_{j=1, j\ne i}^{n}a_{ij}x_j^{(k)})/a_{ii}$$ or in an even more compact form as $$\mathbf{x}^{(k+1)}= \mathbf{D}^{-1}(\mathbf{b}-(\mathbf{L}+\mathbf{U})\mathbf{x}^{(k)}),$$ with $\mathbf{A}=\mathbf{D}+\mathbf{U}+\mathbf{L}$ and $\mathbf{D}$ being a diagonal matrix, $\mathbf{U}$ an upper triangular matrix and $\mathbf{L}$ a lower triangular matrix.