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I am ready to agree beforehand that this looks more like a math.SE question. I posted it there a week ago without any feedback (except for 27 views and 2 upvotes).

Besides, I really need an answer. More precisely, I need a convenient diagonalizability criterion, better than computing the minimal polynomial: for matrices with symbolic entries it is, I believe, useless. Or not?

OK, the question.

In a very nice paper "When Is a Linear Operator Diagonalizable?" by Marco Abate (Amer. Math. Monthly 104 (1997), 824-830) I found the following nice description of the minimal polynomial $\mu(T)$ of an operator $T:V\to V$: for a vector $v\in V$ denote by $\mu_v(T)$, the minimal polynomial of $T$ at $v$ as the smallest possible $a_0+a_1t+...+a_jt^j$ such that $a_0v+a_1T(v)+...+a_jT^j(v)=0$. Then $\mu(T)$ is the least common multiple of $\mu_{b_1}(T)$, ..., $\mu_{b_n}(T)$ for any basis $b_1$, ..., $b_n$ of $V$.

I wonder if this can be used to decide for diagonalizability of $T$ in an efficient way. More precisely, I mean this: we know that $T$ is diagonalizable if and only if the discriminant $\gcd(\mu(T),\mu'(T))$ is nonzero. Can we then utilize discriminants (or maybe some other combinations) of $\mu_{b_i}(T)$ for a basis of our choice, avoiding computation of $\mu(T)$ itself, to decide whether $T$ is diagonalizable?

In other words, is there an efficient way to compute the discriminant of a least common multiple of polynomials in terms of these polynomials, without computing the least common multiple itself?

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  • $\begingroup$ Isn't it true that $\operatorname{lcm}(\mu_1,\dotsc,\mu_n)$ has a repeated linear factor if and only if one of the $\mu_i$ has a repeated linear factor? (I'm assuming your polynomials have coefficients in a closed field, so repeated linear factors is the only thing you have to worry about.) $\endgroup$
    – Zach Teitler
    Commented Jun 9, 2018 at 17:23
  • $\begingroup$ For the record: MSE post is at math.stackexchange.com/questions/2805404/… $\endgroup$
    – Zach Teitler
    Commented Jun 9, 2018 at 17:25
  • $\begingroup$ @ZachTeitler Maybe. However I would prefer to have something that also works for symbolic matrices - say, over fields of rational functions. Of course the thing you say might be useful in this case too: it would mean you just have to check discriminants of the $\mu_i$ separately... $\endgroup$
    – მამუკა ჯიბლაძე
    Commented Jun 9, 2018 at 18:19

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