Good point! Won't necessarily be able to choose the minimiser/maximiser of the ratio

There is actually another proof here, on MSE: math.stackexchange.com/questions/2330328/…. Reading that, I now remember having seen it in the past. Tbh, it's kind of the same idea as the maximum principle that I outline below. But, it's more direct, not going via ranks of matrices

I added a few details to Step 3, for the benefit of anyone else reading this. Hopefully you're happy with them? Step 1 was simple enough that I just left it as it was