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Suppose a linear system is given $$AX=B,$$ where $A\in\mathbb{R}^{n\times n}$ is a symmetric strictly diagonal matrix, and $X, B\in\mathbb{R}^{n\times 2}$. Therefore, the 2D Jacobi iterative solver is applicable here, $$X_{k+1}=D^{-1}(B-RX_k),$$ where the splitting corresponds to $A=D+R$, with $D$ containing only the diagonal entries of $A$. Could it be proved that the iterands $X_0, X_1, X_2, \dots$ are maintaining the line per each row of $X$, meaning that, for $x^i\in\mathbb{R}^2$ being the $i-$th row of $X$, the rows in sequence $x^i_0, x^i_1, x^i_2, \dots$, are colinear (on one line)?

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This was cross posted at scicomp: http://scicomp.stackexchange.com/questions/1558/2d-jacobi-line-maintenance –  Victor Liu Mar 7 '12 at 20:12
    
Yes; I thought about migrating it, but did not manage to do so. Perhaps certain linking by moderators? –  user506901 Mar 8 '12 at 7:47

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