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.