In this paper, we consider the regularized version of the Jacobi algorithm, a block coordinate descent method for convex optimization with an objective function consisting of the sum of a differentiable function and a block-separable function. Under certain regularity assumptions on the objective function, this algorithm has been shown to satisfy the so-called sufficient decrease condition, and consequently, to converge in objective function value. In this paper, we revisit the convergence analysis of the regularized Jacobi algorithm and show that it also converges in iterates under very mild conditions on the objective function. Moreover, we establish conditions under which the algorithm achieves a linear convergence rate.