If $A,B$ are upper triangular matrices such that $AX=XA\implies BX=XB$ for upper triangular $X$, is $B$ a polynomial in $A$?

Question

A professor of mine told me that this is true, but he doesn't remember what the proof was or where to find it, and I haven't been able to find a source for it yet. As such I am looking for one here.

In the theorem as stated, $\mathbb{F}$ is any field and $T_n(\mathbb{F})$ denotes the algebra of upper triangular $n\times n$ matrices over $\mathbb{F}$.

Theorem: Let $A,B\in T_n(\mathbb{F})$ be such that for all $X\in T_n(\mathbb{F})$, $$AX=XA\implies BX=XB$$ Then $B=p(A)$ for some $p\in \mathbb{F}[t]$.

Does anyone know of a source for this result? I have searched Google, MSE, MO, and the like to no avail.

If we replace $T_n(\mathbb{F})$ by $M_n(\mathbb{F})$, the question is answered in this paper. Unfortunately, the argument doesn't seem to translate directly, as I can't find a way to force the $M_i$ maps to be upper-triangular.

Also, I have already asked this question here on MSE. As the question is for an undergraduate research project, it felt appropriate to ask it here as well.

Thanks for any help!

Edit on 9 July, 2018: It's probably worth mentioning that the following theorem is false, so an appeal to Jordan form won't work (at least, not as easily as we'd hope it would).

Fake Theorem: If $A\in T_n(\mathbb{F})$, then there exists an invertible $T\in T_n(\mathbb{F})$ and a permutation matrix $P$ such that $P^{-1}T^{-1} ATP$ is in Jordan form.

An explicit counterexample is $$A=\left[\begin{array}{cccc} 0&1&0&0\\ &0&0&1\\ & &0&1\\ & & &0\end{array}\right]$$ and a more detailed demolishing of this theorem is given here, where the authors prove that if $n\geq 12$ and $\mathbb{F}$ is infinite, then there are infinite sets of nilpotent matrices in $T_n(\mathbb{F})$, none of which are conjugate (in $T_n(\mathbb{F})$) to any of the others.

I mention this because I thought it was true for longer than I'd like to admit, and a few other people I've talked to thought it was true as well until told otherwise.

Answer 1

This is false! Let $$A = \begin{bmatrix} 0&0&0&1 \\ &0&1&0 \\ &&0&0 \\ &&&0 \\ \end{bmatrix}.$$ Imposing that $XA=AX$ for upper triangular $X$ gives linear equations on the $10$ entries of $X$. Solving them, I get that this occurs precisely for $X$ of the form $$X=\begin{bmatrix} a&0&\ast&\ast \\ &b&\ast&\ast \\ &&b&0 \\ &&& a \\ \end{bmatrix}.$$ In turn, an upper triangular matrix commutes with all such $X$ if and only if it is of the form $$B=\begin{bmatrix} c&0&0&d \\ &c&e&0 \\ &&c&0 \\ &&&c \\ \end{bmatrix}.$$ But such a $B$ is only a polynomial in $A$ if $d=e$.

Answer 2

(An observation, not an answer.) If A has pairwise distinct diagonal elements then any matrix $X$ commuting with $A$ is necessarily upper triangular. It is easy to see this once you compute the lower part of $D=[A,X]$, because $$D_{ij}=(A_{ii}-A_{jj})X_{ij}+\sum_{k>i} A_{ik} X_{kj}-\sum_{k<j} A_{kj} X_{ik}.$$ For example, $D_{n1}=(A_{nn}-A_{11})X_{n1}$. Thus $X_{n1}=0$ and then $X_{ij}=0$ by induction on $n-(i-j)$, which only breaks down when $i-j=0$. As $B$ commutes with any matrix commuting with $A$, it is a polynomial of $A$.

What we need to prove is $$C(C({\mathbb F}[A])\cap T_n({\mathbb F}))\cap T_n({\mathbb F})=C(C({\mathbb F}[A])),$$ where $C(R)$ denotes a centralizer of a subalgebra $R\subset M_n({\mathbb F})$. In the above case it is trivial because $C({\mathbb F}[A])\subset T_n({\mathbb F})$.

Answer 3

If $A$ has pairwise distinct diagonal elements one can prove the theorem by induction as follows:

1. By block partitioning one can put $A\in T_{n}({\mathbb F})$ to the form $$ A = \begin{pmatrix} \alpha & a \left(\alpha I - A'\right)\\ 0 & A'\end{pmatrix}$$ with $\alpha \in{\mathbb F}$, $a\in {\mathbb F}^{n-1}$ a row vector and $A'\in T_{n-1}({\mathbb F})$. Here one uses that $\alpha I - A'$ is non singular. For $p \in {\mathbb F}[t]$ one has $$ p(A) = \begin{pmatrix} p(\alpha) & a \left(p(\alpha) I - p(A')\right)\\ 0 & p(A')\end{pmatrix}.$$
2. For $X \in T_{n}({\mathbb F})$, one has $[A,X]=0$ iff $X$ can be put to the form $$ X = \begin{pmatrix} \xi & a \left(\xi I - X'\right)\\ 0 & X'\end{pmatrix}$$ with $[A',X']=0$.
3. Now, if $B\in T_{n}({\mathbb F})$ satisfies $[A,X] = 0\Rightarrow [B,X]=0$ it must be (by Step 2) of the form $$ B = \begin{pmatrix} \beta & a \left(\beta I - B'\right)\\ 0 & B'\end{pmatrix},$$ where $[A',X'] = 0\Rightarrow [B',X']=0$. By induction hypothesis, there is a polynomial $q\in {\mathbb F}[t]$ such that $B'=q(A')$.
4. Now, if one denotes the pairwise distinct diagonal elements (i.e., eigenvalues) of the upper triangular matrix $A$ by $\alpha, \lambda_1,\ldots,\lambda_{n-1}$ the polynomial $p \in {\mathbb F}[t]$ solving the interpolation problem $p(\alpha)=\beta$ and $p(\lambda_j)=q(\lambda_j)$ ($j=1,\ldots,n-1$) yields $p(A')=q(A') = B'$ and hence, by Step 1, $p(A)=B$.

In particular, the thus constructed polynomial $p$ has degree at most $n-1$.

Answer 4

This is not quite the answer, but:

In their paper

Britnell, John R.; Wildon, Mark, On types of matrices and centralizers of matrices and permutations, J. Group Theory 17, No. 5, 875-887 (2014). ZBL1302.15018.

The authors (the paper is available without a paywall)give a characterization of matrices with the same centralizers (note that the OP's question asks about two matrices $A$ and $B$ such that the centralizer of one is a subgroup of a centralizer of the other, but this is not too far. In particular, see Theorem 4.3 and Lemma 5.1 The latter says that two nilpotent matrices have the same centralizer if and only if they are conjugate, which confuses me somewhat, since this does not seem to imply the statement about one being a polynomial in the other (we get to nilpotent if we restrict the OP's situation to unipotent matrices).