Jordan normal form of sum of two commuting nilpotent matrices over a finite field (variant on a linear matrix pencil problem)

Question

This question comes up with trying to construct Lie subalgebras of (large) Lie algebras that are invariant under a finite group $H$. I have two isomorphic $H$-invariant nilpotent subalgebras and am interested in the Jordan normal forms of matrices in diagonal subalgebras of these algebras.

I have two commuting nilpotent matrices $A$ and $B$, (dimension 1596, so cannot be just looked at), defined over the field $\mathbb{F}_9$. They both cube to zero, and so $A+\lambda B$ cubes to zero for any $\lambda\in\overline{\mathbb{F}_3}$. I'm interested in the Jordan normal form of the matrix $A+\lambda B$, where $\lambda$ is a parameter.

In all the examples I have so far, if $A$ and $B$ have the same normal form (in the particular case I have in front of me, blocks $3^{285},1^{741}$) then for all but finitely many values of $\lambda$ the blocks of the sum are the same. Furthermore, the number of exceptions to this statement is small, say around $2$.

This could be because my matrices, coming from Lie algebras, are very special. What I really want to know if the following:

• Is it true that $A+\lambda B$ has Jordan normal form independent of $\lambda$ for cofinitely many $\lambda$?

• Is there a bound on the number of exceptions, say in characteristic $3$ with cube zero matrices?

• If $A$ and $B$ are defined over $\mathbb{F}_q$ then do the exceptions lie in a fixed overfield, say $\mathbb{F}_{q^6}$? (I am thinking $6$ because then all quadratics and cubics in $\lambda$ split. I know that one needs at least $\mathbb{F}_{q^2}$ by examples.)

I really want to know that the JNF of $A+\lambda B$ is what I think it should be for most elements of the algebraic closure, leaving only a finite number to check with a computer. I can do finitely many checks, but not infinitely many! Or is there an algorithm that allows us to understand such problems?

Answer 1

I have since found a way to do this, at least in Magma. (Sage should also be able to do it.) Without Ben Grossmann's way of looking at things I definitely would not have thought of doing this, so I thank him. (This is crucial for my research, so I'm very happy!)

Your pencil looks like $A+xB$ for $x$ a variable. Magma (and Sage) is happy to take echelon forms of matrices over a univariate polynomial ring, so let's do that. I've found it might be better to take a Jordan normal form for $B$ first (and of course conjugate $A$) so as to reduce the number of $x$s in the matrix $A+xB$.

Construct the echelon form of $A+xB$. (For $1596 \times 1596$ matrices over $\mathbb{F}_9$, this takes about a minute for my examples). Then take transposes and take echelon form again.

We now have a matrix with zero off the leading diagonal. Take the multiset of diagonal entries. First, the generic rank of $A+xB$, i.e., the rank for almost all values of $x$, is the number of non-zero entries. Second, the points where this is not the rank are given by the zeroes of the polynomials in the set.

If one needs the full Jordan normal form, one now takes $(A+xB)^2$ and so on, uses the recipe above to compute the rank, check that the exceptional set should be a subset of the exceptional set for $A+xB$, and continue taking powers until you obtain the zero matrix.

Sage, but not Magma, is happy to compute multivariate echelon forms, which is now my next case. Can this be done with non-linear pencils? The set of exceptions is now a variety, one assumes, and things will get significantly more complicated.

Answer 2

Let $n$ denote the size of the matrices $A,B$. Note that the Jordan normal form of a size-$n$ matrix $M$ can be completely recovered if one knows the rank of $(M - t I)^k$ for all eigenvalues $t$ of $M$ and $k = 1,2,\dots,n$ (via the "Weyr Characteristic"). In our case, $A + \lambda B$ has $0$ as its lone eigenvalue, so it suffices to consider the rank of $(A + \lambda B)^k$.

For $k = 1,\dots,n$, let $r_k = \max_{\lambda \in \bar{\Bbb F_3}} \operatorname{rank}(A + \lambda B)^k$. We note that the set $\{M : \operatorname{rank}(M^k) < r_k\}$ is a solution set to a system of polynomials. In particular, it is the set of matrices for which all $r_k \times r_k$ minors are zero.

This is enough for us to deduce that for each $k$, we have $\operatorname{rank}(A + \lambda B)^k = r_k$ for cofinitely many $\lambda$. In particular, the common zero set to a system of polynomials is the same as the zero-set of the product of these polynomials, and a zero-set of a polynomial of one variable must either be the entirety of $\Bbb {\bar F}$ or a finite subset.

Thus, $(A - \lambda B)$ must have constant Jordan form (corresponding to the maximal ranks $r_k$) for cofinitely many $\lambda \in \bar{\Bbb F_3}$.

Moreover, it is possible to obtain a bound on the number of solutions by considering the number and degrees of equations attained by setting the appropriate minors to $0$.

If we know that there are at most $m$ exceptions, then we also know that these exceptions are the zeros of a polynomial with degree at most $m$ and coefficients in $\Bbb F_q$, but this polynomial must split over $\Bbb F_{q^m}$. That is, a positive answer to your second question implies a positive answer to your third.

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To be a bit more specific: we know that $A + \lambda B$ cubes to zero for all $\lambda$. With that, the only two equations that need to be accounted for are $$ \operatorname{rank}(A + \lambda B) < r_1, \quad \operatorname{rank}(A + \lambda B)^2 < r_2. $$ We have $\binom n r^2$ size-$r$ minors, and the entries of $(A + \lambda B)^2$ have at most degree $2$ with respect to $\lambda$. So, the first inequality gives us a system of $\binom n{r_1}$ degree-$r_1$ equations. Similarly, the second inequality gives us a system of $\binom n{r_2}$ degree-$2r_2$ equations. This is enough to deduce that the solution set in question is necessarily the zero set of some polynomial with degree $$ m \leq 2\binom n{r_1}\binom n{r_2}r_1r_2. $$ Note that if $\operatorname{rank}(A + \lambda B) = r_1$, $\operatorname{rank}(A + \lambda B)^2 = r_2$, and $(A + \lambda B)^3 = 0$, then we know that the Jordan form has $n-r_1$ blocks in total, with $r_1 - r_2$ blocks of size at least $2$ and $r_2$ blocks of size at least $3$. Because $(A + \lambda B)^3 = 0$, the Jordan form has no blocks of size $4$ or greater. All together, this gives us $n + r_2 - 2r_1$ blocks of size $1$, $r_1 - 2r_2$ blocks of size $2$, and $r_2$ blocks of size $3$.

If there are no blocks of size $2$, then we have $r_1 = 2r_2$, and $r_2$ blocks of size $3$.

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We could get a better bound on the number of exceptions as follows. Because $B$ has no blocks of size $2$ we find that (up to similarity over $\Bbb F_q$) $B$ must have the form $$ B = \pmatrix{0 & I_{k} & 0\\ 0 & 0 & I_{k}\\ 0 &0 &0\\ &&&0}, $$ with $k = \frac 12 \operatorname{rank}(B)$. Because $A$ and $B$ commute, $A$ must have the form $$ A = \pmatrix{A_1 & A_2 & A_3 & A_{14}\\0 & A_1 & A_2 & 0\\ 0 & 0 & A_1 & 0\\ 0&0&A_{43}& A_{44}}, $$ So, that $$ A + \lambda B = \pmatrix{A_1 & A_2 + \lambda I_{k} & A_3 & A_{14}\\ 0 & A_1 & A_2 + \lambda I_k & 0\\ 0 & 0 & A_1 & 0\\ 0&0&A_{43}& A_{44}}. $$ Now, note that the degree of a size $k$ minor for $A + \lambda B$ is at most equal to the minimum of the number of rows and the number of columns selected that correspond to either the $1,2$ or $2,3$ block.

Note that if $SA_1S^{-1}$ is in Jordan form, then we have $$ \pmatrix{S \\ & S \\ && S\\ &&& I}(A + \lambda B) \pmatrix{S \\ & S \\ && S\\ & &&I}^{-1} = \\ \pmatrix{SA_1S^{-1} & SA_2S^{-1} + \lambda I_{k} & SA_3S^{-1}\\ 0 & SA_1S^{-1} & SA_2S^{-1} + \lambda I_k\\ 0 & 0 & SA_1S^{-1} \\ &&& A_{44}}. $$ Similarly, we can also put $A_{44}$ into its Jordan form without losing any structure.