Given the matrix $A_{M,N}$ with $N\gt M$, the vector $y$, I have to find the vectors $x$ and $u$, satisfying the following equations: $$D(x)x=A^Tu$$ $$y=Ax$$ where: $$D(x) = \left| \begin{array}{ccc} \exp(-x_1^2/2\sigma^2) & 0 & ... \\ 0 & \exp(-x_2^2/2\sigma^2)& ... \\ ... & ... & ...\\0 &0&\exp(-x_M^2/2\sigma^2) \end{array} \right| $$ The vector $x$ is: $$x=\left( \begin{array}{ccc} x_1 \\ ... \\ x_M \end{array} \right)$$ in which $x_1,x_2,...x_M$ are the unknown variables. The vector $u$ is: $$u=\left( \begin{array}{ccc} u_1 \\ ... \\ u_N \end{array} \right)$$ Is there some way to solve it? Thanks in advance.
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$\begingroup$ Why do you have to do this? $\endgroup$– Fernando MuroCommented Jul 17, 2014 at 15:17
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$\begingroup$ @FernandoMuro: It's a problem arising in the field of signal processing. I need to solve it before to proceed with other calculation. $\endgroup$– Riccardo.AlestraCommented Jul 17, 2014 at 15:18
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$\begingroup$ So $A$ is actually $N \times M$? And did you really want $N > M$, or the reverse? $\endgroup$– Robert IsraelCommented Jul 18, 2014 at 18:55
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$\begingroup$ @RobertIsrael: I want $M\gt N$ $\endgroup$– Riccardo.AlestraCommented Jul 21, 2014 at 8:18
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