The [[Jacobi Method]] is a type of [[Iterative Method]] to find the solution to a [[System of Linear Equations]]. For a given [[System of Linear Equations]] represented in matrix form, $ \huge A\vec{x} =\vec{ b} $ Where $A$ is an $n\times n$ [[Matrix]]. Let us notate the value of the $n$-th iteration as $\mathbf{\vec x}^{(n)}$. We start with an arbitrary initial iteration $\mathbf{\vec x}^{(1)}$. Each following iteration $k+1$ is given by the following: $ \huge \begin{align} \mathbf{x}_{i}^{(k+1)} &= \frac{1}{a_{ii}} \pa{ b_{i} - \sum_{j \neq 1}a_{ij}x_{j}^{(k)} } \end{align} $ Where $i=0,1,\dots, n$. Each next iteration $\mathbf{\vec x}^{(k+1)}$ is solved as an [[Affine Transformations|Affine Transformation]] of the previous, $\mathbf{\vec x}^{(k)}$. For the solution $\vec x$, let $\vec y^{(k)}= \mathbf{\vec x^{(k)}}-\vec x$. If this method convergences, then the limit of $\vec{y}$ will converge to the [[Zero Vector]]. $ \huge \lim_{ k \to \infty } \vec y^{(k)} = \vec 0 $