The question I am going to ask probably has a straight-forward answer, I just don't know how to frame my question in a way that would be 'normal.'
I have a set of a set of linear equations that I would like to do a regression on. The main equation will always look like the standard matrix linear equation system:
$Ax = b$
where A is a 3x3 matrix, x is 3x1 and b is 3x1. However, I can gather data to make 6 equations of this form and A should be the same for each one.
$Ax_1 = b_1 \\ Ax_2 = b_2 \\ Ax_3 = b_3 \\ Ax_4 = b_4 \\ Ax_5 = b_5 \\ Ax_6 = b_6$
Please note that $x_1$, $x_2$, $b_1$, $b_2$, etc. are not single values, but are 3x1 vectors. So for a given matrix A, I can produce 6 sets of x's ad b's. Because I am gathering data from sensors that will have noise, I don't want to just solve $Ax_1 = b_1$ and throw away the data I can get from the other 5 systems. Is there a way to do a linear regression not on a system of linear equations, but on a system of a system of linear equations?