# Numerical algorithm for finding unknown values of non square matrix

Suppose I have a non-square matrix where some rows of values are known, and others are not.

$$\begin{bmatrix}x_{11}&x_{12}&x_{13}\\x_{21}&x_{22}&x_{23}\\ y_{31}&y_{32}&y_{33}\\x_{41}&x_{42}&x_{43}\\\end{bmatrix}$$

Assume that all X values are the known values (the entire rows' values will be known) and that there are rows where all the values are unknown, the Y values.

We also know a few other key points of information. We know the linear algebra equation comes out to XV = W where V is a vector of variables $$\begin{bmatrix}v_1\\v_2\\v_3 \end{bmatrix}$$ and W is a vector that is the dot product of X and V. All values of W are known.

For example, the row of Y values, while we don't know the individual Y values, we know that $$y_{31}*v_1 + y_{32}*v_2 + y_{33}*v_3 = w_3$$ and we actually know the true value of $$w_3$$, just not the $$y$$ values that lead to it.

Is there a way to solve for possible values of the unknown $$y$$ values? Looking to understand the linear algebra behind the concept and will eventually imploy solution (if possible) in python.

• So you have a matrix with nine given entries, and you have one linear equation involving the other three entries? Then no, you can't determine all three of them from that.
– Ian
Apr 22 '20 at 18:59

You can't extract those values. Suppose $$y_{31} v_1 + y_{32}v_2 + y_{33} v_3 = w_3$$. Then $$(y_{31} + v_2)v_1 + (y_{32} -v_2)v_1 + y_{33}v_3 = w_3$$ also. So if at least two of your $$v_j$$ are nonzero, you will certainly not have a unique solution.
If you get to pick your vector $$v$$. Just take $$v = (1,0,0)$$, then $$Xv = y_{31}$$, $$v=(0,1,0)$$, gives you $$Xv= y_{32}$$ and so on.