The real non-singular matrix $A \in \mathbb{R}^{m \times m}$ is written in the form $A=A_{D}+A_{U}+A_{L}$, where the matrices $A_{D}, A_{U}, A_{L} \in \mathbb{R}^{m \times m}$ are diagonal and non-singular, strictly uppertriangular and strictly lower-triangular respectively.

Given $b \in \mathbb{R}^{m}$, the Jacobi iteration for solving $A x=b$ is

$$A_{D} x_{n}=-\left(A_{U}+A_{L}\right) x_{n-1}+b, \quad n=1,2 \ldots$$

where the $n$th iterate is $x_{n} \in \mathbb{R}^{m}$. Show that the iteration converges to the solution $x$ of $A x=b$, independent of the starting choice $x_{0}$, if and only if the spectral radius $\rho(H)$ of the matrix $H=-A_{D}^{-1}\left(A_{U}+A_{L}\right)$ is less than 1 .

Hence find the range of values of the real number $\mu$ for which the iteration will converge when

$$A=\left[\begin{array}{rrr} 1 & 0 & -\mu \\ -\mu & 3 & -\mu \\ -4 \mu & 0 & 4 \end{array}\right]$$