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Suppose that I have this over-determined system of equations,

$$a_1x_1 + a_2x_2 + a_3x_3 = k_1$$ $$b_1x_1 + b_2x_2 + b_3x_3 = k_2$$ $$c_1x_1 + c_2x_2 + c_3x_3 = k_3$$ $$d_1x_1 + d_2x_2 + d_3x_3 = k_4$$

Normally, if this system of equation can be constructed in matrix multiplication form as $$\begin{bmatrix} a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \\ c_1 & c_2 & c_3 \\ d_1 & d_2 & d_3 \end{bmatrix}\begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix} = \begin{bmatrix} k_1 \\ k_2 \\ k_3 \\ k_4 \end{bmatrix}$$

And this can be solved in a least-square sense. However, I have found in MATLAB that if I repeated the first equations for $n$ times, i.e.

$$\begin{bmatrix} a_1 & a_2 & a_3 \\ a_1 & a_2 & a_3 \\ a_1 & a_2 & a_3 \\ \vdots & \vdots & \vdots \\ a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \\ c_1 & c_2 & c_3 \\ d_1 & d_2 & d_3 \end{bmatrix}\begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix} = \begin{bmatrix} k_1 \\ k_1 \\ k_1 \\ \vdots \\ k_1 \\ k_2 \\ k_3 \\ k_4 \end{bmatrix}$$

The answers to this system equation would not be the same, when I solved it by using moore-penrose pseudoinverse. My question is that if we extract a system of equations from this matrix multiplication, we still get the same set of equations as before. And why do we get different answers?

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  • $\begingroup$ What do you mean by: the answers to this system equation are not the same?? $\endgroup$ – Martigan May 13 '15 at 16:39
  • $\begingroup$ Sorry for an unclear statement. I meant that when I solved this matrix equation by using moore-penros pseudoinverse, it did not return the same answer as before. $\endgroup$ – aofkrittin May 13 '15 at 16:41
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    $\begingroup$ The pseudo-inverse are obviously different. But when you solved the equations, you got some different results for $x_i$? $\endgroup$ – Martigan May 13 '15 at 16:46
  • $\begingroup$ So you are saying that, if the system is more likely to have repeated equations, it is not safe to use pseudo-inverse in order to solve in least square sense. Am I right? $\endgroup$ – aofkrittin May 13 '15 at 16:49
  • $\begingroup$ I have one more question. What is the implication of putting repeated equations in the system? Does that mean I am weighting each equation? $\endgroup$ – aofkrittin May 19 '15 at 10:18

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