The Non-negative Inverse Eigenvalue Problem for Tridiagonal Matrix, Circulant Matrix and Symmetric Matrix

The Non-negative Inverse Eigenvalue Problem (NIEP) is a sub-problem extracted from inverse eigenvalue problem with a long history from 1930s determining the sufficient and necessary conditions in order that, σ={ λ 1,…, λ n }to be the spectrum of an entry wise non-negative n x n matrix. There had been many excellent researchers and scholars who contributed to discover many practical theories. The following of this paper would be then separated into two parts to further analyze the NIEP. In the first section of the paper, some important preexisting conclusions would be demonstrated and the groups’ understanding of these indispensable theories would be expressed. In the second section, three special and wildly used matrix, including tridiagongal matrix, circulant matrix, and symmetric matrix would be considered. The solution of the NIEP of these matrices done by the group would be expressed.

Study of necessary conditions of non-negative inverse eigenvalue problem, this so called inverse problem, is the study of conditions which eigenvalues of a non-negative matrix should satisfy, which is: Similarly, if we expand polynomial Compare the coefficient of correspondence, we have: i.e. ∑ 0 Then, we talk about ) ( N k A k  . Recall that over C we can always change bases so that we achieve diagonal representation. This is to say that for any n n matrix A, we can find a transition matrix X to get AX X B 1   , which is a diagonal matrix and the new bases  is the spectrum of k A , which is non-negative too. (k-th moment) ⋯ 0, ∈ According to Perron-Frobenius theorem [4], the spectral radius of a nonnegative matrix must, itself, be an eigenvalue. Therefore, there must be a non-negative one, at least as big in absolute value as any others. Without loss of generality, we let 1  be the non-negative one, then we have: The last one is the famous JLL condition [5,6]: (JLL) , , 1,2, ⋯ is the non-negative matrix we are looking for Necessity.
We assume that 2 1    , then we have 0 0 We launch the contradiction.
In fact, when n=2, Both of them are necessary conditions as we mentioned in 1.2, but now, when n=2, they become necessary sufficient and necessary condition.
The above discussion is just an example, actually, researchers have drawn more advanced conclusions.
Theorem (     According to the definition, second order non-negative tridiagonal matrix is just a non-negative matrix without any other special properties. Therefore, we will talk about the NIEP from third order non-negative tridiagonal matrix.
We assume that According to Vieta's formulas, For simplicity, let: According to (3), let's talk about the value of , , .When 0, 0, but we already knew that 0. It doesn't make sense. So that 0.Similarly, we know that 0, 0.Then, we want to work out , , by simultaneous equations (2) Let (4)-(5), we have However, we already know that It's impossible.
Focusing on the numerator，we find that in equation (6) 0 In equation (7) 0  However, since the sign of is unknown, things are different from situation 1), it is hard for us to determine the sign of numerator and we can't launch contradiction anymore. So we try to find a solution of equations (6), (7). Having failed over and over again, we finally find a proper solution. We let According to equations(1),(8), we find a solution for them Here, we let Here, we let 0 2 . It can be inferred that the actual values of We have discussed all situations, then we can conclude that Theorem (2.1.3). For a proposed spectrum , the sufficient and necessary condition under which they can be realized by a (symmetric) non-negative tridiagonal matrix is (a) Obviously, it is not complete to discuss only the real numbers. What if the spectrum is composed with a real number and a complex conjugate?
We assume that   ,the necessary condition for it to be achieved by a (symmetric) non-negative tridiagonal matrix is At this stage, we want to conclude about the methods in the above discussion. The method of using vieta's formula to represent the elements in the non-negative tridiagonal matrix and then determine the sign of the elements to find contradiction when researching the necessary conditions of the NIEP is quiet It is circulant matrix.
Step one: We refer to the simplest form of this kind of matrices, which we name it 1 With K, we can construct any circulant matrix with addition and multiplication. For example,   Step two: For any unit cycle matrix K, which is also the simplest form of circulant matrices, we can calculate it's eigenvalue through this equation , where  refers to the eigenvalue of matrix K. It's easy to get that It is the transformation matrix of K. In other words,

Theorem (2.2.2)
The sufficient and necessary condition, under which ) ( C is n-dimensional circulant matrix, is that Step three: We can just solve the inverse eigenvalue problem of circulant matrix by discussing the first line or row of it, thus we can get every element of circulant matrix. Specifically, we assume that the matrix we need is

Symmetric matrix
We already know and look for a symmetric matrix A whose eigenvalue is . The transformation matrix of A is P. Based on properties of eigenvalue, we get an Here, we introduce a property of symmetric matrices. Noticing that is not only a diagonal matrix but also a symmetric matrix as well, it applies to this property.

Lemma(2.3.1)
If X is a symmetric matrix, then for any matrix B, T BXB is also a symmetric matrix. In this case, for any matrix B, T B B is a symmetric matrix. Based on these, in order to find a symmetric matrix A through eigenvalue, T P must be equal to 1

Definition(2.3.2)
If a matrix P satisfies

Conclusion
Although the non-negative inverse eigenvalue problem (NIEP) is a long-standing problem and corresponding researching tools are limited, in our research, you will find that when the matrix is special matrix, such as tridiagonal matrix, circulant matrix and symmetric matrix, and the dimension is limited, this long-standing puzzle is entirely solvable.

The non-negative inverse eigenvalue problem for tridiagonal matrix
In this paper, 3-dimension non-negative tridiagonal matrix is considered. According to the Vieta's formulas, we build a bridge between the eigenvalue and the elements of the matrix. While finding a proper solution, we separate the eigenvalue into different groups, based on the different construction of spectrum. On the one hand, in some situations, we find the corresponding matrix, which means this kind of eigenvalue can be achieved by non-negative tridiagonal matrix, on the other hand, sometimes we get a paradox, which means this kind of eigenvalue isn't realizable. However, unfortunately, when the eigenvalue is composed by a real number and a pair of conjugate complex numbers, we can neither find a suitable solution nor derive a contradiction.

The inverse eigenvalue problem for cyclic matrix
For circular matrix, in this paper, we start with the most basic unit cyclic matrix. We solve the inverse eigenvalue problem of unit cyclic matrix at first, then simplifying the general cyclic matrix inverse eigenvalue problem base on that general cyclic matrix can be written as a combination of unit matrix. In this way we find the transformation matrix F and get a solution for general cyclic matrix inverse eigenvalue problem. When it comes to the inverse eigenvalue problem of cyclic matrix, the uniqueness of the obtained cyclic matrix remains to be studied. For example, under what circumstances the obtained cyclic matrix C is unique?
3.3. The inverse eigenvalue problem for symmetric matrix In this paper, by using the special characters of symmetric matrices, the conditions that transformation matrices should satisfy are obtained. Thus a general solution to the inverse eigenvalue problem of symmetric matrices is derived.

Expectation and application
Furthermore, based on our research, it's possible to establish an R package or set up a function in MATLAB. When entering a set of eigenvalues and the desired matrix category( like tridiagonal matrix, circulant matrix, and symmetric), the corresponding matrix can be output. However, when the input eigenvalues do not meet the corresponding necessary conditions, it shows not found. In short, our theory can be realized by computer language, which is convenient for the following researchers to verify and calculate.