Symmetric Integer Matrices Having Integer Eigenvalues

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.


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 Z 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][4][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 {0, 1, . . .,  − 1}; 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   () = det( − ) is called the characteristic polynomial of .It is clear that the characteristic polynomial is an th degree polynomial in  and det( − ) = 0 will have  (not necessarily distinct) solutions for .The values of  that satisfy det( − ) = 0 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   (Z) be the set of all  ×  symmetric matrices with integer entries.
We now present the characterization of all symmetric integer matrices for rank at most 2 that have integer spectrum.

The Rank 2 Case
Theorem 3. Let  ∈   (Z) with rank 2. Then  has integer eigenvalues if and only if there exist two integers  and  such that trace() =  +  and  [2] =  where  [2] 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 2 , 2 . ( "⇐" Suppose there exist integers  and  such that trace() =  +  and  [2] = .Then it follows that "⇒" If all eigenvalues of  are integers, then there exists an integer  such that trace () Letting and using (7), the difference of trace() 2 and  2 is 4 [2] which is even, so either both trace() and  are even or both trace() and  are odd, and hence both  and  are integers.
Then we can write  as Since  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.

The Rank 3 Case
Theorem 7. Let  ∈   (Z) be symmetric integer matrix with rank 3.If one of the following cases holds, then  has integer eigenvalues.
(i) One of the eigenvalues of  is 1 or −1 and there exists a positive integer  such that (ii) All nonzero eigenvalues of  are the same and (iii) One of the nonzero eigenvalues of  has multiplicity two and there exists a positive integer such that (iv) 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  = 1.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 1 − trace and  are even or both of them are odd by (20).
(ii) Let λ be the only nonzero eigenvalue of .Then By comparing the coefficients on both sides of (24), one obtains trace () = 3 λ, Thus (iii) Suppose that  has two nonzero eigenvalues  1 and  2 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  1 has multiplicity two, both   ( 1 ) and its derivative    ( 1 ) 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  1 , 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  1 ,  2 , and  3 .We have  3 = −( 1 +  2 ) due to zero trace.Also, we have the following equations Multiplying (34) by  1 yields that Subtracting ( 35) from (36), one obtains Note that the following is always a solution of (36).To see this, first note that 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].