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This might be a dumb question, it's been a long time since I've done any linear algebra.

I have an $M_{m\times n}$ matrix, who's values are the sum of a row and column vectors $\vec{x}_{1\times n}$ and $\vec{y}_{m\times 1}$. eg:

$M=\begin{bmatrix} x_1+y_1&x_2+y_1 &\cdots& x_n+y_1 \\ x_1+y_2&x_2+y_2 &\cdots& x_n+y_2 \\ \vdots & \vdots & \ddots & \vdots \\ x_1+y_m & x_2+y_m & \cdots & x_n+y_m \end{bmatrix}$

If I know the values of $M$, how do I solve for $x$ and $y$?

I'm sure there's a name for this, but I can't remember.

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  • $\begingroup$ Are you given the matrix in the form of your example? As in, does the matrix specify the sum of the column entries and two entries? $\endgroup$
    – Seeker
    Oct 26, 2022 at 20:30
  • $\begingroup$ I only have the values in M, but the relationship is known. As a concrete example if I have $\begin{bmatrix} 0.9 & -0.8 \\ 1.6 & -0.1 \end{bmatrix}$, what are $x_1,x_2,y_1,y_2$? Do I even have enough information to solve this? $\endgroup$
    – brent.reynolds
    Oct 26, 2022 at 20:49
  • $\begingroup$ I can’t think of a way to solve this. But wait around and see if someone comes up with a way. Is there any other additional details you have. Context can really help in solving a problem. $\endgroup$
    – Seeker
    Oct 26, 2022 at 20:54
  • $\begingroup$ The larger problem I'm trying to solve has to do with collecting measurements, where each measurement has a contribution from 2 independent components. This is the $x$ & $y$ in my example. My thought was that if I can average all the data for each $x_i$ & $y_i$ I could populate the above matrix, and then solve for $x$ & $y$. $\endgroup$
    – brent.reynolds
    Oct 26, 2022 at 21:09
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    $\begingroup$ to me it seems a system of $nm$ equations in $n+m$ unknowns, which is overdetermined if $nm>n+m$. $\endgroup$
    – l4teLearner
    Oct 26, 2022 at 21:33

1 Answer 1

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I don't think this is solvable in all honesty.

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