Things that make sense
Nov. 5th, 2008 09:40 pm![[personal profile]](https://www.dreamwidth.org/img/silk/identity/user.png){kind=small}
sniffnoySo I only realized a few days ago that in a Euclidean domain you can use Euclidean algorithm to do a sort of pseudo-Gauss-Jordan. (Is there a name for it? I'll just call it Euclidean Gauss-Jordan for now.) So useful! Had I realized it a few days earlier, I would have been able to complete problem 6 on that week's algebra homework.
Also in algebra the other day I finally learned how to compute rational canonical form algorithmically! Well, not exactly; Ginzburg didn't directly talk about that at all. See in Dummit & Foote, in addition to their main proof of finitely generated modules over PIDs, they give an additional, algorithmic proof, just for Euclidean domains, which in particular gives you how to compute rational canonical forms algorithmically. However, they leave the key step of the algorithm as an exercise - one I hadn't been able to figure out. Ginzburg, with the sloppiness he has displayed before, attempted to do this proof in class (though it was modified to work for general PIDs rather than Euclidean domains), took the same approach I did, but couldn't quite make it work either. Fortunately, he came in the next day with a correct version - which, though not itself algorithmic, was easy enough to deduce an algorithm from (in the PID case, assuming you know a norm for your domain satisfying N(a,b)<N(a) for b∉(a)). So, yay, I finally know how to do that.
-Harry
Also in algebra the other day I finally learned how to compute rational canonical form algorithmically! Well, not exactly; Ginzburg didn't directly talk about that at all. See in Dummit & Foote, in addition to their main proof of finitely generated modules over PIDs, they give an additional, algorithmic proof, just for Euclidean domains, which in particular gives you how to compute rational canonical forms algorithmically. However, they leave the key step of the algorithm as an exercise - one I hadn't been able to figure out. Ginzburg, with the sloppiness he has displayed before, attempted to do this proof in class (though it was modified to work for general PIDs rather than Euclidean domains), took the same approach I did, but couldn't quite make it work either. Fortunately, he came in the next day with a correct version - which, though not itself algorithmic, was easy enough to deduce an algorithm from (in the PID case, assuming you know a norm for your domain satisfying N(a,b)<N(a) for b∉(a)). So, yay, I finally know how to do that.
-Harry
