Jacobi method- convergence.

Question

I'm trying to find positive-definite matrix $A$ (3x3), such that Jacobi method is not convergent.

Answer 1

A simple example is $$A =\begin{pmatrix} 1.5 & 1 & 1 \\ 1 & 1.5 & 1 \\ 1 & 1 & 1.5 \end{pmatrix} $$ with eigenvalues $\sigma(A) = \{0.5, 3.5\}$ thus the matrix is (symmetric) positive definite. But the spectral radius of the error propagation matrix is $$\rho \big(I-D^{-1} A\big) = \bigg \vert 1 - \frac{1}{1.5} \cdot 3.5 \bigg \vert = \bigg \vert 1 - \frac73 \bigg \vert = \frac43 > 1$$ and thus the method is not convergent (for every starting vector).