2 Lengths of families of rank-one matrices

One recent article of Longstaff [Reference Longstaff2] is mostly devoted to the following question.

Theorem 3.5 of [Reference Longstaff2] shows $\ell (F)\leqslant n$ . Longstaff gives examples with $\ell (F)\in \{1,2\}$ and $\ell (F)\in \{n-1,n\}$ , and also $\ell (F)=k+1$ , for any positive integer k dividing $n-1$ (see [Reference Longstaff2]). Despite considerable attention and citations to the work [Reference Longstaff2], no further progress seems to have been made, and Question 2.1 was unsolved.

The solution is as follows. As in the article of Longstaff [Reference Longstaff2], we focus on families consisting of matrix units $e_{ij}$ (that is, each $e_{ij}$ is the matrix having $1$ at the position $(i,j)$ and zeros everywhere else). We recall that a simple graph G has finite sets of vertices V and edges E, and each edge is a two-element subset of V. A path between $v_1,v_2\in V$ is a list of edges $(e_1,e_2,\ldots ,e_k)$ with $v_1\in e_1$ and $v_2\in e_k$ , and also $|e_i\cap e_{i+1}|=1$ , for all $i\in \{1,2,\ldots ,k-1\}$ . A graph is connected if any pair of vertices $u\neq v$ allows a path between them, and the distance between u and v is the length of the shortest path. The diameter of a connected graph G is the largest possible distance between any two distinct vertices in G (see [Reference Ore5]).

The set $F=\{(1,2),\ldots ,(n-1,n),(n,1)\}$ has $\ell (F)=n$ , as in [Reference Longstaff2], and graphs with n vertices and diameters $1,2,\ldots ,n-1$ are straightforward (see Figure 1 and [Reference Ore5]).

Figure 1

Complete graphs with at least two vertices have diameter one. The right graph has n vertices and diameter $d\geqslant 2$ .

Therefore, the set of all possible values of $\ell (F)$ in Question 2.1 is $\{1,2,\ldots ,n\}$ for all $n\neq 1$ , giving a complete answer to Question 2.1.

3 Lower bounds on slot length

The article [Reference Longstaff3] is devoted to the concept of slot lengths of matrix families.

Definition 3.1 [Reference Longstaff3, page 260]

The slot length of a family $S\subset \mathrm {Mat}_n(\mathbb {C})$ is the smallest integer k such that the algebra generated by S is spanned by the elements

$$ \begin{align*}(s_1)^{\varepsilon_1}\cdot(s_2)^{\varepsilon_2} \cdots (s_k)^{\varepsilon_k}\end{align*} $$

involving arbitrary elements $s_i\in S$ taken to arbitrary powers $\varepsilon _i\geqslant 0$ .

Longstaff shows that the slot length of an irreducible pair $\{A, B\}\subset \mathrm {Mat}_n(\mathbb {C})$ is at most $n-1$ , provided that A is a Jordan cell and $n\geqslant 5$ , (see [Reference Longstaff3, Theorem 2.7]). The following potential generalisation was proposed in further discussion in [Reference Longstaff3].

However, this conjecture is false. Indeed, let $n=2k+1$ be an odd number and J the nilpotent Jordan cell of order n. We take

$$ \begin{align*}A=J^{k} \quad\text{ and }\quad B=(J^\top)^{k+1}\end{align*} $$

with $J^{\top }$ being the transpose of J. We immediately have

(3.1) $$ \begin{align} A^3=B^2=0 \end{align} $$

and $\operatorname {rank} A^2=1$ , which shows that, for all matrices $U, V, W\in \mathrm {Mat}_n(\mathbb {C})$ ,

(3.2) $$ \begin{align} U\cdot A^2\cdot V\cdot A^2\cdot W\quad\text{ is a scalar multiple of }\quad U\cdot A^2\cdot W. \end{align} $$

Finally, we recall that Laffey, Markova and Šmigoc [Reference Laffey, Markova and Šmigoc1, Theorem 2.12] proved that

(3.3) $$ \begin{align} \ell(\{A, B\})=2n-2, \end{align} $$

and that $\{A, B\}$ is irreducible [Reference Laffey, Markova and Šmigoc1, Theorem 2.10]. If the slot length of $\{A, B\}$ is less than $2n-3$ , then the algebra $\mathrm {Mat}_n(\mathbb {C})$ is spanned by products of the forms

$$ \begin{align*}A^{\varepsilon_1}\cdot B^{\varepsilon_2}\cdot A^{\varepsilon_3} \cdots A^{\varepsilon_{2n-5}}\cdot B^{\varepsilon_{2n-4}} \quad\text{ and }\quad B^{\varepsilon_1}\cdot A^{\varepsilon_2}\cdot B^{\varepsilon_3} \cdots B^{\varepsilon_{2n-5}}\cdot A^{\varepsilon_{2n-4}}\end{align*} $$

with any exponents $\varepsilon _i\geqslant 0$ , as in Definition 3.1. By the conditions (3.1) and (3.2), the algebra $\mathrm {Mat}_n(\mathbb {C})$ is now spanned by products of A and B of lengths not exceeding $2n-3$ , which is a contradiction to (3.3). Therefore, the slot length of $\{A,B\}$ cannot be less than $2n-3$ , providing a counterexample to Conjecture 3.2.

4 Irreducible sets of three square-zero matrices

This section is a remark on one other result in [Reference Longstaff4]. Following [Reference Longstaff4], a square matrix A is called a square-zero matrix if $A^2=0$ . In [Reference Longstaff4, Proposition 2.3], Longstaff gives a quite complicated demonstration of the following fact.

However, this is easy to see. Indeed, for $\chi \in \{0,1\}$ , one can define $M_\chi $ as the sum of matrix units $e_{i\,i+1}$ over all those indices $i\in \{1,2,\ldots ,n-1\}$ which are of the same parity as $\chi $ . As the sum $M_0+M_1$ is the nilpotent Jordan cell J, we can show that the family of three square-zero matrices $\{M_0,M_1,e_{n1}\}$ is irreducible. Indeed,

$$ \begin{align*}e_{ab}=J^{n-a}\cdot e_{n1}\cdot J^{b-1}\end{align*} $$

is an expression of all matrix units (see also [Reference Laffey, Markova and Šmigoc1, Theorem 2.10]).

5 Upper bounds on $1$ -minimum spanning lengths

Let $S\subset \mathrm {Mat}_n(\mathbb {C})$ be a finite set and $x=(x_1,x_2,\ldots ,x_t)$ a word with $x_i\in S$ . This word is said to have height one if, in each consecutive pair, the two letters are different, that is, $x_i\neq x_{i+1}$ for $i\in \{1,2,\ldots ,t-1\}$ (see [Reference Longstaff4, page 78]). If, for some integer k, the set of all products of nonempty height-one words,

(5.1) $$ \begin{align} x_1\cdot x_2 \cdots x_t \end{align} $$

with $t\leqslant k$ , spans the full matrix algebra $\mathrm {Mat}_n(\mathbb {C})$ , Longstaff defines the $1$ -minimum spanning length $\operatorname {msl}_1(S)$ as the smallest k with this property [Reference Longstaff4, page 78].

Finally, the minimum spanning length $\operatorname {msl}(S)$ is defined in the same way as $\operatorname {msl}_1(S)$ , except that the words (5.1) are not required to have height one [Reference Longstaff4, page 78].

In [Reference Longstaff4, Remark 2.4], Longstaff conjectures the inequality $\operatorname {msl}_1(\{A,B\})\leqslant 4n-7$ for any irreducible pair $\{A,B\}\subset \mathrm {Mat}_n(\mathbb {C})$ with $n\geqslant 2$ . As also noted in [Reference Longstaff4, page 80], the Cayley–Hamilton theorem does not allow the height-one words in A and B to span a subspace of dimension greater than $4n-2$ . This number is less than $n^2$ whenever $n\geqslant 4$ , and then the conjecture becomes void. In the context of [Reference Longstaff4, Remark 2.4 and Proposition 2.5], a possible intended meaning could have been that

(5.2) $$ \begin{align} \operatorname{msl}(\{A,B\})\leqslant 2n-2\quad \text{implies}\quad \operatorname{msl}_1(\{I,A,B\})\leqslant 4n-7 \end{align} $$

for irreducible pairs $\{A,B\}\subset \mathrm {Mat}_n(\mathbb {C})$ . Indeed, [Reference Longstaff4, Remark 2.4] gives a weaker version of (5.2), with $4n-7$ replaced by $4n-5$ . Another approach to (5.2) is [Reference Longstaff4, Proposition 2.5], but it involves an additional assumption on $\{A,B\}$ , as follows:

(5.3) $$ \begin{align} \text{ both } A^{n-1}\,B^{n-1} \text{ and } B^{n-1}\,A^{n-1} \text{ are in } V_{2n-3}(A, B)+W(A, B), \end{align} $$

where $V_j(A, B)$ is the linear span of the nonempty products of length at most j of $\{A, B\}$ , and $W(A, B)$ is the span of all products of those words of length $2n-2$ that are different from both $A^{n-1}\,B^{n-1}$ and $B^{n-1}\,A^{n-1}$ .

For $n=2$ , the conjecture (5.2) can be refuted with the irreducible pair $\{e_{12}, e_{21}\}$ and easy calculation. However, it turns out that the conjecture is always true with $n\geqslant 3$ . In other words, the requirement of (5.3) is not necessary in Lemma 5.2.

Now, arguments similar to [Reference Longstaff4, Proposition 2.5] complete the proof of (5.2).

Proof. Let $\mathcal {B}$ be a basis of $\mathrm {Mat}_n(\mathbb {C})$ consisting of products of lengths at most $2n-2$ . Using the Cayley–Hamilton theorem, we can assume that no product of $\mathcal {B}$ has n consecutive appearances of either A or B. Further, if

(5.4) $$ \begin{align} A^{n-1}\,B^{n-1}\notin\mathcal{B} \quad\text{ and }\quad B^{n-1}\,A^{n-1}\notin\mathcal{B}, \end{align} $$

then we insert the multiplication by the identity I between every pair of consecutive equal letters in the words of $\mathcal {B}$ . This results in a basis of height-one words at the cost of adding at most $2n-5$ letters in each word, and hence, $\operatorname {msl}_1(\{I,A,B\})\leqslant 4n-7$ . If (5.4) is false, then either $A^{n-1}\,B^{n-1}\in \mathcal {B}$ or $B^{n-1}\,A^{n-1}\in \mathcal {B}$ . Now, we write

$$ \begin{align*}A^{n-1}\,B^{n-1}=(A^{n-1}B)\cdot B^{n-2} \quad\text{ and }\quad B^{n-1}\,\,A^{n-1}=(B^{n-1}A)\cdot A^{n-2}\end{align*} $$

to further replace the $A^{n-1}B$ and $B^{n-1}A$ parts by words of $V_{n-1}(A, B)$ , $W'(A, B)$ and $W'(B, A)$ in Lemma 5.3. Thus, each of the resulting words is in the linear span of $V_{2n-3}(A, B)$ and those products of length $2n-2$ that have at least two pairs of distinct consecutive letters, which reduces the situation to the known case (5.4).