Disproving a conjecture of Thornton on Bohemian matrices

1 Introduction

A matrix is called Bohemian if its entries are of a bounded height, typically drawn from a discrete set. The term “Bohemian” is intended as a mnemonic and derived from “BOunded HEight Matrix of Integers.” These matrices are of relevant interest in many areas where their spectral properties are relevant. The consideration to this type of properties goes back to more than one century ago, as pointed out in [1,2]. For a more global historical perspective and current trends, the reader is referred to [3,4,5] and references therein.

Meanwhile, the study of these matrices has become independent and, based on numerous and hard computational experiments, many conjectures have emerged in the literature. The most important collection can be found in the so-called Characteristic Polynomial Database [6].

Quite recently, this catalogue attracted the attention of several researchers, namely on those conjectures concerned to Hessenberg matrices. Indeed, they are extended and proved in [7,8,9]. This claim has two exceptions though. One of them is solved and generalized in [10], the remaining unsolved one is [6, Conjecture 9].

We recall that the first 11 terms of sequence [11, A212264] are listed in Table 1, where we understand A 21226 4 n as the n th term of the sequence A212264.

In this paper, we will show that, for n = 10 , this conjecture is false. Next, we analyze the Hessenberg matrices aforementioned in detail. Then we establish several results which will lead to a counterexample, concluding that the conjecture is false. In Section 5, we provide a theoretical approach for such evidence. Moreover, we claim that the conjecture is also false for n = 11 . More generally, our methods are valid for any n as well.

2 Preliminaries

In this section, we analyze the Hessenberg matrices mentioned in Conjecture 1 in detail. We recall that these matrices have relevant properties and applications (cf., e.g., [12,13,14, 15,16]).

Let ℳ n be the set of all n × n upper-Hessenberg matrices with entries from the set { 0 , 1 } , subdiagonal entries all equal to 1, and diagonal entries all equal to 0. That is to say, each matrix in ℳ n is of the following form:

where ∗ takes either the value 0 or 1. Several open problems and results involving these matrices can be found, for example, in [17,18].

Let us denote by a n the number of distinct determinants among all matrices in ℳ n . Conjecture 1 claims that { a n } n ⩾ 0 agrees with A212264.

It is easy to see that, after expanding the determinant of the matrix defined in (2.1) along the first column and the last row, one can obtain:

Based on this observation, we introduce a modified version of ℳ n , say N n . The only difference between ℳ n and N n lies on the first and last diagonal entries, which are 0 in ℳ n , but 1 in N n . Hence, each matrix in N n is of the following form:

A close relation about the determinants of matrices in ℳ n and N n is revealed by (2.2). So, the investigation about the determinants of the matrices in N n can assist us during the study of Conjecture 1.

3 Matrices in N n

Let DN n be the set of all possible determinants of matrices in N n and set b n = ∣ DN n ∣ .

The next corollary is an immediate consequence of the previous lemma.

4 A counterexample

For any integers a and b with a ⩽ b , we use [ a , b ] to represent the set of all integers between a and b , that is, { a , a + 1 , … , b − 1 , b } . The first few values of DN n and b n up to n = 8 can be found in Table 2 (we will introduce how to obtain them in Section 5).

Taking into account the relation a n = b n − 2 , for n ⩾ 5 , presented in Lemma 3, we obtain the first few values of a n up to n = 10 immediately (Table 3). Note that the values of a_n with n up to 4 are obtained directly.

If we compare now the values of a n with A212264 shown in Tables 1 and 3, respectively, one can see that they agree when 0 ⩽ n ⩽ 9 , but not when n = 10 . This evidence indicates that Conjecture 1 is true for small n ( 0 ⩽ n ⩽ 9 ), but false when n = 10 . Consequently, we may state

5 A theoretical analysis

In this section, we present how to obtain the data in Table 2 theoretically.

Write A ∈ N n as

The expansion of the determinant of A , for n ⩾ 4 , along the second column, leads to

Let us consider the determinants of the two matrices in (5.1) separately.

We start looking at the first matrix. If a 34 = 1 , then this matrix is in N n − 3 . Otherwise, we go ahead with further expansion(s) for the first column with only one 1, until the first diagonal entry is equal to 1 or a matrix of order 1 is left. This implies that the resulting matrix should be in N s , with 1 ⩽ s ⩽ n − 3 , or it is the matrix of order 1 whose unique entry is equal to 0.

As to the second matrix, it belongs to N n − 1 when a 23 = 0 . Otherwise, we have a 23 = 1 and, under such condition, we may assume that a 13 = 0 (if a 13 = 1 , denote by C i the i th column of A , after applying the column elementary transformation C 3 − C 1 to A , the ( 2 , 3 ) -entry becomes 0, which is equivalent to the case when a 23 = 0 ), interchanging the first two rows of the second matrix leads to a matrix in N n − 1 again, but the determinant of this new matrix and that of the initial one are symmetric about the origin.

Based on the above arguments, we may decompose det A into two parts as

where x ∈ DN s , with 1 ⩽ s ⩽ n − 3 , or x = 0, and y ∈ DN n − 1 .

Now we are ready to obtain DN n up to n = 8 as claimed in Table 2. It is trivial to get DN n for small n : DN 0 = [ 1 , 1 ] , DN 1 = [ 1 , 1 ] , DN 2 = [ 0 , 1 ] , and DN 3 = [ − 2 , 1 ] . Suppose now that n = 4 . From (5.2), s can only take the value 1, x ∈ DN 1 = [ 1 , 1 ] , or x = 0, and y ∈ DN 3 = [ − 2 , 1 ] ( ± y ∈ [ − 2 , 2 ] ), it is not hard to see that det A ∈ [ − 2 , 3 ] from (5.2) (take the union of all possible values based on all combinations of x and y ). Similarly, we can deduce that:

• det A ∈ [ − 4 , 4 ] , when n = 5 ;

• det A ∈ [ − 6 , 5 ] , when n = 6 ;

• det A ∈ [ − 8 , 9 ] , when n = 7 ;

• det A ∈ [ − 13 , 13 ] , when n = 8 .

At this stage, we find the possible intervals of det A , for any A ∈ N n , up to n = 8 . Of course, this is not enough to obtain DN n . In fact, we still need to search for matrices in N n whose determinants cover all the integers in those assumed intervals.

For 4 ⩽ n ⩽ 7 , DN n is indeed the corresponding intervals obtained above (the way for constructing the desired matrices can be derived from the case n = 8 below). However, when n = 8 , [ − 13 , 13 ] is not the right interval. In fact, DN 8 should be [ − 12 , 12 ] . Let us illustrate how to construct matrices such that their determinants attain any given integer in [ − 12 , 12 ] (this method is also valid for 4 ⩽ n ⩽ 7 ), and then explain why ± 13 cannot be the determinants of matrices in DN 8 .

Observe that DN 7 = [ − 8 , 9 ] . From the proof of Lemma 1 and Remark 3.1, we may construct matrices whose determinants are [ − 9 , 9 ] . Aiming to extend this interval to [ − 12 , 12 ] , let us use the matrix in N 7 with (extremal) determinant 9:

It is worth mentioning that this matrix is the unique one in N 7 with determinant 9, which can be verified by a systematic use of the construction we are introducing, or the algorithm we present in Section 6 (see b 7 , 9 = 1 in Table 4).

Using the construction introduced in the proof of Lemma 1 and Remark 3.1, we may construct matrices in N 8 with determinants − 9 and 9, respectively:

and

Next, taking some entries in the third row as variables, we get the determinants

and

After choosing suitable values for a , b , c , d , e , we can obtain matrices whose determinants are in [ − 12 , − 10 ] or [ 10 , 12 ] .

In order to make the conclusion that DN 8 = [ − 12 , 12 ] , we still need to show that det A ≠ ± 13 , for any A ∈ N 8 . From (5.2), if det A = 13 for some matrix A ∈ N 8 , then a 12 = 1 , s = 5 , x = 4 , ± y = 9 , i.e., A is of the form shown in (5.7) (because after expanding along the second column, one of the obtained matrices has order 7 and determinant − 9 ), but its maximal determinant is just 12. Similarly, if det A = − 13 , then A is of the form shown in (5.6), and its minimal determinant is just − 12 .

In conclusion, DN 8 = [ − 12 , 12 ] , and thus b 8 = a 10 = 25 . So Conjecture 1 is false when n = 10 .

Similarly, we may get det A ∈ [ − 18 , 17 ] for A ∈ N 9 ( DN 9 is not necessarily [ − 18 , 17 ] , may be it is just a subset of [ − 18 , 17 ] ), so b 9 = a 11 ⩽ 36 , i.e., Conjecture 1 is also false when n = 11 (since A 21226 4 11 = 37 ).

6 The distribution of determinants for matrices in N n

Let b n , k be the number of matrices in N n whose determinants are k . Clearly,

We list all the b n , k for 2 ⩽ n ⩽ 8 and any possible k in Table 4. These values are obtained by Algorithms 1 and 2 outlined below.

From the structure of matrices in N n (see (2.3)), if the matrix is of order n , then there are n ( n − 1 ) 2 unknown entries (all of which lie above the main diagonal), taking 0 or 1. It is natural that we may associate these n ( n − 1 ) 2 values with a binary bit array of cardinality n ( n − 1 ) 2 according to the lexicographic ordering. For example, in a matrix A = ( a i j ) of order 3, the unknown entries are a 12 , a 13 , a 23 listed in the lexicographic ordering, we would assign a 12 ≔ A [ 0 ] , a 13 ≔ A [ 1 ] , a 23 ≔ A [ 2 ] according to Algorithm 1. It provides us a way to construct a one-to-one correspondence between matrices in N n and the integers ranging from 0 to 2 n ( n − 1 ) 2 − 1 .

Based on the aforementioned one-to-one correspondence, we are ready to enumerate the number of matrices in N n sharing the same determinants (i.e., generate b n , k ). This can be realized by using Algorithm 2, where we use C [ n ] to represent the determinants of matrices in N n , in terms of the binary bit array A = [ A [ 0 ] , A [ 1 ] , … , A [ 2 n ( n − 1 ) 2 − 1 ] ] produced by Algorithm 1. With the help of MATLAB, we obtain all the C [ n ] , for 2 ⩽ n ⩽ 8 (Table 5).