Here is yet another definition in mathematics where it seems that conventions vary (almost like the orientation of titles on the spines of books): is a Jordan block an upper-triangular or lower-triangular matrix? In other words, which of the matrices
is a Jordan block of size 2 with respect to the eigenvalue ?
I have the vague impression that most elementary textbooks in Germany (I taught linear algebra last year…) use , but for instance Bourbaki (Algèbre, chapitre VII, page 34, définition 3, in the French edition) uses , and so does Lang’s “Algebra”. Is it then a cultural dichotomy (again, like spines of books)?
I have to admit that I tend towards myself, because I find it much easier to remember a good model for a Jordan block: over the field , take the vector space , and consider the linear map defined by . Then the matrix of with respect to the basis is the Jordan block in its lower-triangular incarnation. The point here (for me) is that passing from to is nicely “inductive”: the formula for the linear map is “independent” of , and the bases for different are also nicely meshed. (In other words, if one finds the Jordan normal form using the classification of modules over principal ideal domains, one is likely to prefer the lower-triangular version that “comes out” more naturally…)