Hello Folks, Am looking for some perspective on the following problem/attempt...I have 2 separate rank-deficient linear system of equations (ie there's a null space to each and they aren't the same null spaces). There's at least some overlap in their column subspaces. Furthermore, the null space of one system also overlaps with the column space of the other (and vice-versa). Theoretically, it would mean that there's some complementary info in each system, which would never show up in the individual solutions (ie solution by solving each system seperately). To capture this "additional" information, I intend combining the 2 system of equations and then solving the single new system. Does it make sense? Is it a futile task? Thanks for your thoughts...

It certainly depends on how you "combine them". If you solve them separately in the LS sense, then you're minimizing ||A1x1 - b1|| and ||A2x2 - b2|| subject to x1, x2 having minimum norm. An immediate thing to do would be to solve [A1 A2][x1;x2] = [b1;b2] minimizing ||Ax1 + Ax2 - b1 - b2|| subject to [x1;x2] having minimum norm. But note that ||Ax1 + Ax2 - b1 - b2|| <= ||Ax1 -b1|| + ||Ax2 - b2||. So the error in the combined system is at least as good as the sum of the two errors separately. Thus, if you're goal is to minimize the overall error then it makes sense to combine the two.