Jordan blocks

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

A_1=\begin{pmatrix}\alpha & 1\\0&\alpha\end{pmatrix},\quad\quad A_2=\begin{pmatrix} \alpha & 0\\1&\alpha\end{pmatrix}

is a Jordan block of size 2 with respect to the eigenvalue \alpha?

I have the vague impression that most elementary textbooks in Germany (I taught linear algebra last year…) use A_1, but for instance Bourbaki (Algèbre, chapitre VII, page 34, définition 3, in the French edition) uses A_2, 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 A_2 myself, because I find it much easier to remember a good model for a Jordan block: over the field K, take the vector space V=K[X]/X^nK[X], and consider the linear map u\colon V\to V defined by u(P)=\alpha P+XP. Then the matrix of u with respect to the basis (1,X,\ldots,X^{n-1}) is the Jordan block in its lower-triangular incarnation. The point here (for me) is that passing from n to n+1 is nicely “inductive”: the formula for the linear map u is “independent” of n, and the bases for different n 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…)

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Kowalski

I am a professor of mathematics at ETH Zürich since 2008.

3 thoughts on “Jordan blocks”

  1. Naturality of your choice depends on wheather there is a culture which would rather write (X^{n-1},…,X,1). Relevant thing, the creation operator is denoted as a^\dag, which makes it secondary with respect to the annihiliation operator a (contrary to my feelings, while I cannot annihilate without creationg first). Here the natural basis is given by natural numbes (up to 1/2 shift) and a is A_1 while a^\dag is A_2 in your notation.

  2. On one hand, for typesetting purposes it is easier to write row vectors, so having matrices act on the right is more natural. There are also many actions (e.g. Galois, group) which we represent on the right. On the other hand, applying a function to an element of its domain is typically a left action: f(g(x))=(fg)(x). (Isaacs book *Algebra: a graduate course* bucks this trend by having functions act on the right. It’s quite convenient since it aligns with the aforementioned right actions, but it goes against most conventions.)

    1. Jacobson’s short book on Lie algebras also has actions on the right, if I remember correctly. It’s indeed confusing when comparing with other books (especially when trying to learn Lie algebras, since issues of sign, etc, also come up, and the translation between right and left is hesitant at first.)

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