# How to find row and column vectors given a matrix of sums.

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.

• 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? Oct 26, 2022 at 20:30
• 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? Oct 26, 2022 at 20:49
• 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. Oct 26, 2022 at 20:54
• 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 $x$ & $y$. Oct 26, 2022 at 21:09
• to me it seems a system of $nm$ equations in $n+m$ unknowns, which is overdetermined if $nm>n+m$. Oct 26, 2022 at 21:33