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When students come to the College (first two years of the University system in most of the developped countries) to train in mathematics, they get a linear algebra / matrix analysis course. After a few months, perhaps after one year, they are taught about diagonalization of matrices. They learn many criteria that are either necessary or sufficient or both. This seems to be a mandatory step for future engineers and other categories of scientific workers.

My question is a bit provocative:

Is diagonalization that important? Should we teach it thoroughly to people who will have to use linear algebra and matrices in the future?

Here are a few arguments why we should refrain ourselves to enter this topic, except when teaching future mathematicians:

1- The solution of this problem is not so nice, many matrices being not diagonalizable. And the set of diagonalizable matrices is neither open nor close in any sense (usual, if the field is $\mathbb R$ or $\mathbb C$, Zariski otherwise).

2- Diagonalization is not effective. As a matter of fact, we cannot compute explicitely the eigenvalues of an $n\times n$ matrix if $n\ge5$ (Abel plus companion matrix).

3- Diagonalization is not really useful. You don't use it to calculate the exponential, or to invert, ... What engineers are interested in is often stability of dynamical systems. Thus a good problem is whether the spectrum belongs to either the left half-plane or the unit disk, whether there are eigenvalues on the unit circle or with vanishing real part.

I therefore open a discussion, in which I am looking for either pro- or con- arguments about teaching diagonalization to engineers.

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    $\begingroup$ This question strikes me as a little too subjective for MO; it would be more appropriate for a blog post (perhaps a guest post on someone else's blog?). $\endgroup$
    – Qiaochu Yuan
    Commented Mar 27, 2012 at 15:47
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    $\begingroup$ I agree with @Qiaochu -- I am not sure what is to be gained from this discussion. I should just note that for engineers, over the complex numbers, all matrices are diagonalizable, and effectively too (just ask Matlab). For mathematicians, the importance of diagonalization is hard to overstate. $\endgroup$
    – Igor Rivin
    Commented Mar 27, 2012 at 15:56
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    $\begingroup$ I agree that MO isn't the right place for this sort of debate. I also don't think it's really about diagonalization per se: I don't see any way to answer that question seriously without addressing the broader questions of who is studying linear algebra, what they plan to do with it, what they need to know to do that, and how much more they should study than they will actually need (to maximize their understanding of what they will need). This is an important discussion, but not well suited to the MO question/answer format. $\endgroup$
    – Henry Cohn
    Commented Mar 27, 2012 at 16:30
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    $\begingroup$ What about these less subjective versions: (1) Are there theorems in linear algebra that can be proved with diagonalization and don't have known proofs without? (2) Are there arguments in linear algebra that are better understood (intuitively) with diagonalization than without? $\endgroup$
    – darij grinberg
    Commented Mar 27, 2012 at 16:47
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    $\begingroup$ @quid: a generic matrix is diagonalizable over $\mathbb{C}$ (since, in particular, a generic polynomial has distinct roots...), so in practice you never see non-diagonal Jordan forms. Computing the eigenvalues of a matrix is a very heavily studied problem, for which very efficient algorithms are known. Again, engineers have absolutely no interest in computing solutions in radicals (for that matter, neither do mathematicians), so the OP's companion matrix comment is not really relevant to any sort of practice. $\endgroup$
    – Igor Rivin
    Commented Mar 27, 2012 at 18:17

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You are talking about training in mathematics, not numerical analysis, so the answer is surely "yes". What is more, given that software packages exist, it should be taught in a more conceptual way than is done traditionally. I'm struck by how much of the "Moscow School" or Gel'fand way of looking at things depends on a good feel for the basics of linear algebra,

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