Ah, i think i understand now, so the reason they have a rivalry between them is because, i THINK, this is in no way official, but it is pretty plausible, that Mil's mother, out of jealousness, love, or desire to be empress, killed Linden's mother and sister, thus driving her mad,
and since she was Mil's mother, he still loves her, despite how crazy she is as of now, and since Linden's "plans" are taking revenge on EVERYONE who caused that incident, this includes primarily Mil's mother, and i'm pretty sure that since Lin loved both her mother and sister quite a lot, revenge means murdering them in this case, but Mil won't let that happen, so long as he's alive that is.