Discussion
Consider the linear system \(A\mathbf{x} = \mathbf{b}\), where \(A = [a_{ij}]\), \(i, j = 1, 2, 3\), and \(a_{ii} \neq 0\) for \(i = 1, 2, 3\), is a matrix with entries in \(\mathbb{R}\). For
\[ D = \begin{bmatrix} a_{11} & 0 & 0 \\ 0 & a_{22} & 0 \\ 0 & 0 & a_{33} \end{bmatrix}, \]
let
\[ D^{-1}A = \begin{bmatrix} 1 & 1 & -2 \\ 3 & 1 & 2 \\ 1 & 1 & 1 \end{bmatrix} \]
and
\[ D^{-1}\mathbf{b} = \begin{bmatrix} 4 \\ 4 \\ 1 \end{bmatrix}. \]
Consider the following two statements:
S1: The approximation of x after one iteration of the Jacobi scheme with initial vector \(\mathbf{x}_{0} = [1\ \ 1\ \ 1]^{T}\) is \(\mathbf{x}_{1} = [5\ \ {-1}\ \ {-1}]^{T}\).
S2: There exists an initial vector \(\mathbf{x}_{0}\) for which Jacobi iterative scheme diverges.
Then, which one of the following is correct?
Answer: C
Both S1 and S2 are TRUE.
The Jacobi scheme for the scaled system solves each equation for its diagonal unknown using the previous values:
\[ \begin{aligned} x_{1} &= 4 - x_{2} + 2x_{3} = 4 - 1 + 2 = 5 \\ x_{2} &= 4 - 3x_{1} - 2x_{3} = 4 - 3 - 2 = -1 \\ x_{3} &= 1 - x_{1} - x_{2} = 1 - 1 - 1 = -1 \end{aligned} \]
so S1 holds. The iteration matrix is \(B = I - D^{-1}A\), whose characteristic equation works out to \(\lambda^{3} - 3\lambda - 4 = 0\). It has a real root near 2.2, so the spectral radius exceeds 1 and the iteration diverges for a general starting vector (any whose error has a component along that eigenvector). So S2 holds too.
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