So, fun MATLAB story. For my thesis I had to do a little root finding method. Basically Newton's Method in many dimensions. So, make a big matrix, compute a big "derivative" matrix, solve Ax=b to find the direction to go. The variables I was solving for were coefficients in a basis expansion (basis elements were spherical harmonics). I was only looking for real solutions, but the solver would actually go off the real axis before converging on a real solution, even with real initial data. In certain situations, there would always be a really weird reflection in some sense that essentially amounted to a sign issue. I couldn't figure out what it was, so I ended up just adding a negative sign to "fix" the problem. It didn't make any sense to me, but the answer was right so I let it sit like that for a while. (I only had to do the computation some of the time.. the bulk of the work was just proofs.)
Finally towards the end of the thesis I went back to it and had to adjust some of the code. I still came across this confusing sign issue. Finally I realized that I was taking matrix transposes with the ' operator (i.e., A' for transpose(A)) because, well, I assumed that was the transpose and it seemed like it was. But, A' is really the conjugate transpose, and A.' is the nonconjugate transpose. I didn't need the conjugate; I only needed to switch rows and columns for the dimensions to correspond correctly. Of course, this made all i's into -i's, and this gave rise to the reflection issue in those situations. I felt pretty silly after I realized it.
(If you're wondering why i -> -i gave rise as a reflection issue, it's because part of the basis functions were of the form e^{m i x} for m = -N, ..., 0 , ..., N. So, flipping i to -i flipped m to -m, which made those basis functions mismatch with the other m's. The solver converged to real solutions, so the complex component was always 0, so when looking at the solution you couldn't really see the i to -i issue.)
tl;dr; I used conjugate transpose instead of transpose (' instead of .') without realizing it and hacked a fix to correct it without discovering the obvious reason for, like, years.
