= Solution
Positive definiteness of $A$ implies $D>0$. Let $B=D^{-1/2}AD^{-1/2}$, 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 \b[convergence from every starting vector].
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