Symmetric Integer Matrices Having Integer Eigenvalues

Abstract

We provide characterization of symmetric integer matrices for rank at most 2 that have integer spectrum and give some constructions for such matrices of rank 3. We also make some connection between Hanlon’s conjecture and integer eigenvalue problem.

1. Introduction

The study of matrices with integer entries combines linear algebra, number theory, and group theory (the study of arithmetic groups). It was shown that the eigenvalues of symmetric matrices over the integers stem from as to what algebraic integers occur as eigenvalues for the incidence matrix of a graph (see [1]). Integer eigenvalues of a nonsymmetric matrix with entries as certain simple functions are presented in [2]. A graph is Laplacian integral if the spectrum of its Laplacian matrix consists entirely of integers. A number of papers on Laplacian matrices investigate the class of Laplacian integral graphs (see [3–5]). Integer matrices that arise from Laplacians are connected to the three-dimensional Heisenberg Lie algebra and the eigenvalues and eigenvectors were explicitly given for the subclass of these matrices (see [6]). An interesting class of matrices called was introduced in [7]; the most interesting property of the -class is that the spectra of the matrices consist of the consecutive integers ; that is, the eigenvalues do not depend on the values of the elements of In this paper, we characterize all symmetric integer matrices for rank at most 2 that have integer spectrum and give some constructions for such matrices of rank 3. We also open a discussion on the fact that integer eigenvalue problem has strong connection with Hanlon’s conjecture (see [6]). We provide some examples and conjectures that relate these two problems.

We start with some basic definitions from linear algebra. Let be a square matrix of size and let be a scalar quantity. Then is called the characteristic polynomial of It is clear that the characteristic polynomial is an th degree polynomial in and will have (not necessarily distinct) solutions for . The values of that satisfy are the characteristic roots or eigenvalues of . An matrix is called real symmetric if , the transpose of , coincide with . If is an matrix and is an matrix, then the tensor product of and , denoted by , is the matrix and is defined as If is and is , then the Kronecker sum (or tensor sum) of and , denoted by , is the matrix of the form Let be the set of all symmetric matrices with integer entries.

Let be an matrix and be the sum of all th order principal minors of Then all coefficients in the characteristic polynomial of can be expressed by for In particular, and

Lemma 1. Let with eigenvalues The characteristic polynomial of is given by Then

Lemma 2. Let with rank 1. Then has integer eigenvalues.

We now present the characterization of all symmetric integer matrices for rank at most 2 that have integer spectrum.

2. The Rank 2 Case

Theorem 3. Let with rank 2. Then has integer eigenvalues if and only if there exist two integers and such that and where is the sum of determinants of all 2nd order principal minors of

Proof. Since has rank 2, the characteristic polynomial of has the form It is clear that the two nonzero eigenvalues of are “” Suppose there exist integers and such that and Then it follows that “” If all eigenvalues of are integers, then there exists an integer such thatLetting and using (7), the difference of and is which is even, so either both and are even or both and are odd, and hence both and are integers.

Lemma 4. Let If is invertible and , then

Theorem 5. Let Suppose both and are integer matrices of order with integer eigenvalues. If and commute, then has integer eigenvalues.

Proof. According to Lemma 4, Since and commute, they can be diagonalized simultaneously and hence all eigenvalues of both and are integers.

Lemma 6. LetIf , then has integer eigenvalues.

Proof. DefineThen we can write asSince and have integer eigenvalues, then has also integer eigenvalues.

We now give some constructions for all symmetric matrices of rank 3 that has integer spectrum.

3. The Rank 3 Case

Theorem 7. Let be symmetric integer matrix with rank 3. If one of the following cases holds, then has integer eigenvalues.One of the eigenvalues of is or and there exists a positive integer such that All nonzero eigenvalues of are the same andOne of the nonzero eigenvalues of has multiplicity two and there exists a positive integer such that The trace of is equal to zero and there exists a positive integer and integers , such that  In fact, one of eigenvalues is

Proof. Since the rank of is 3, the characteristics polynomial of can be written as(i)Suppose that one of the eigenvalues of is By substituting this eigenvalue in (17), one obtains In addition, (17) can be factored as By Theorem 3, the quadratic factor has integer roots if and only if there exists a positive integer , such that Now combining (18) and (20) yields And in fact, the other eigenvalues are  which are integers because either both and are even or both of them are odd by (20).(ii)Let be the only nonzero eigenvalue of . Then  By comparing the coefficients on both sides of (24), one obtains Thus (iii)Suppose that has two nonzero eigenvalues and with multiplicity one and two, respectively. Then the characteristic polynomial of can be written as  and hence By comparing the coefficients on both sides of (28), one obtains In addition, since has multiplicity two, both and its derivative are equal to zero and hence Now taking the derivative of both sides of (28), we get Since (31) is quadratic and has one integer root , then the other root must be rational. Thus there exists a positive integer such that which yields that(iv)Denote the nonzero eigenvalues of by , , and We have due to zero trace. Also, we have the following equations Multiplying (34) by yields that Subtracting (35) from (36), one obtains Note that the following is always a solution of (36). To see this, first note that

Lemma 8. Letand let Then has integer eigenvalues if and both and have integer eigenvalues.

Proof. Since and commute, they can be diagonalized simultaneously. Without loss of generality, suppose Note that has eigenvalues with multiplicity for and has eigenvalue with multiplicities for since has rank

Lemma 9. Let If where each has integer eigenvalues and , , then has integer eigenvalues.

Proposition 10. Let and Suppose both and have integer eigenvalues. Then has integer eigenvalues.

In this section, we open a discussion on possible connection between integer eigenvalue problem and Hanlon’s conjecture. We support our approach with some examples and adopt the notation used in [6].

4. Connection to Hanlon’s Conjecture

Definition 11. Let , , and be nonnegative integers with and Let be the set of pairs such that is an -subset of and is a -subset of , where and is a nonnegative integer. Define the weight of a pair to be Let be the set of pairs such that

Example 12. Let , , and If , then the ordered basis is If , the ordered basis is In this case, if then

We define a matrix with respect to corresponding basis We need the following definition in order to define the matrix

Definition 13. Let and be elements of , and let be a triple with , and is any integer. we say that and are -neighbors if(1),(2),(3).

In [6] (Conjecture ), it was conjectured that the eigenvalues are nonnegative integers. Let us present some examples.

Example 14. with respect to the ordered basis is Theorem 3 guarantees that both eigenvalues of are integers.

Example 15. with respect to the ordered basis isIf the last row and column are deleted, the resulting matrix is in the form of block matrix given in Theorem 5. The eigenvalues of -block are 1, 3 and the eigenvalues of -block are both 1 with multiplicity 2. The eigenvalues of are . Notice that these eigenvalues can also be obtained from the sum and difference of and blocks with given multiplicities.

Example 16. with respect to the ordered basis isThe eigenvalues of are , , , , , and . We do not observe an obvious connection in this example.

Example 17. with respect to the ordered basis is

Example 18. with respect to the ordered basis isNotice that the matrix is in block form given in Theorem 5 and its eigenvalues are the eigenvalues of the sum and difference of and -blocks with given multiplicities. In this case the eigenvalues are , , , and

Example 19. with respect to the ordered basisis

Note that some examples given above follow directly from the theory we have provided and some do not. We aim to search more general connections in a followup paper. In [6], the author search for algebraic expression for the eigenvalues. We use the same notation and state the conjecture. For each , , , , and each nonnegative integer , let denote the multiplicity of as an eigenvalue of Let

Let be the following generating function for the numbers :

It was conjectured in [6] (Conjecture ) that This conjecture is still open in general. We hope to find more obvious connections and use them to solve given conjectures in this paper.

Competing Interests

Lei Cao and Selcuk Koyuncu declare that there is no conflict of interests regarding the publication of this paper.