 Solution

Positive definiteness of $A$ implies $D>0$. Let $B=D^{-1/2}A{D}^{-1/2}$, a symmetric positive definite matrix. The additional hypothesis says $2I-B=D^{-1/2}(2D-A)D^{-1/2}>0$, so all eigenvalues $\lambda$ of $B$ satisfy $0<\lambda <2$. The ordinary Jacobi error matrix $I-D^{-1}A$ is similar to $I-B$ and therefore has eigenvalues $1-\lambda \in (-1,1)$. Its spectral radius is less than one. The convergence criterion for stationary linear iterations consequently gives convergence from every starting vector.