Diagonal realizability in the Nonnegative Inverse Eigenvalue Problem
We show that if a list of nonzero complex numbers is the nonzero spectrum of a diagonalizable nonnegative matrix, then is the nonzero spectrum of a diagonalizable nonnegative matrix of order
The nonnegative inverse eigenvalue problem (NIEP) asks which lists of complex numbers can be the spectrum of some entry-wise nonnegative matrix. If a list of complex numbers is the spectrum of some entry-wise nonnegative matrix , we say that is realizable, and that realises . The NIEP is a difficult open problem, however, several partial results are known. For other sources of literature on the problem we refer the reader to the following works and the citations that appear in them: [3, 5, 8, 9, 12, 13, 14].
Motivated by applications in ergodic theory, Boyle and Handelman  solved a related question: which lists of complex numbers can be the nonzero spectrum of a nonnegative matrix? In particular, they proved that if is a list of complex numbers such that the power sums for all positive integers and for , then there exists a nonnegative integer such that the list obtained by appending zeros to is realizable by a nonnegative matrix. It can be shown that the least required here is in general not bounded as a function of The proof in  is not constructive and does not enable one to determine the size of the required for realizability in the general case. A constructive approach to the Boyle and Handelman result that provides a bound on , the number of zeros needed for realizability, was given by Laffey in .
Several other variants of the NIEP have been considered in the literature. One that attracted a lot of attention is the symmetric nonnegative inverse eigenvalue problem (SNIEP), where it is demanded that the realising nonnegative matrix is symmetric. The corresponding question about the nonzero spectrum of a symmetric matrix is open. Unlike in the general case, the number of zeros needed to be added to the nonzero spectrum of a symmetric nonnegative matrix in order to obtain a nonnegative symmetric realisation is bounded by a function of the number of nonzero elements in the the list.
Theorem 1.1 ()
Let be a symmetric nonnegative matrix of rank . Then, there exists a symmetric nonnegative matrix with the same nonzero spectrum as .
This result was used in the first proof that showed the symmetric nonnegative inverse eigenvalue problem is different to the real nonnegative inverse eigenvalue problem, for the problem of determining which lists of real numbers are realizable. The bound provided in the theorem above is believed not to be tight. In fact, examples of lists where one zero added makes the list symmetrically realizable are known, but at present there are no known examples where three or more zeros are required in symmetric realizability.
In this note, we consider analogous questions for diagonal realizability. In particular, we show that if a list is the nonzero spectrum of a diagonalizable nonnegative matrix with nonzero eigenvalues, then it can be realised by a nonnegative diagonalizable matrix of order
The ideas that we use in this note are similar to those in , where Carathéodory’s theorem plays a central role.
Theorem 1.2 (Carathéodory)
Let be an -dimensional vector space, and let . Let be the convex cone generated by . Then each point in can be expressed as a linear combination, with nonnegative coefficients, of or fewer of the ’s.
2 Main Results
Our approach will depend on the existence of a principal sub-matrix of the original matrix that has the same rank as . We start by considering the structure of a matrix with a principal submatrix of the same rank as .
Let and have the same rank. Then there exists an matrix such that
and is similar to
Proof. Since , there exists an matrix so that , and since is equal to the rank of:
we conclude that . This gives us:
to verify the second part of the statement.
Our first application of this lemma given below considers the general case.
be a nonnegative matrix, where has rank equal to the rank of and the rank of is equal to . Furthermore, we assume .
Then there exists a nonnegative matrix whose nonzero spectrum is the same as the nonzero spectrum of . Moreover, the Jordan canonical forms of and , denoted by and respectively, satisfy: .
Proof. By Lemma 2.1
hence the nonzero spectrum of is equal to the nonzero spectrum of
Let us write down the columns of and the rows of
Let denote the span of ’s, and let denote its orthogonal complement. The vector space
has dimension , and contains , for By Theorem 1.2 we can write:
where We define:
Note that is obtained from by deleting some rows and corresponding columns of , and then multiplying some of the surviving columns by nonnegative constants. Hence, the nonnegativity of is clear from the construction. Furthermore, the nonzero spectrum of is equal to the nonzero spectrum of , since
We still need to prove the connection between the Jordan forms of and . From the construction of we see that , but since contains a principal submatrix whose rank is the same as rank of , we conclude that .
Lemma 2.1 tells us that is similar to
and is similar to
We use a permutation similarity on to deduce that is similar to a matrix of the form:
Since , the rows of are linear combinations of rows of and . Hence we can find an matrix and an matrix such that a similarity of the form:
on the matrix results in:
Now we know that is similar to a matrix of the form and the relationship between and follows.
The following lemma allows us to obtain a bound on the size of in terms of the rank of and the number of nonzero eigenvalues of in the results above.
Let have nonzero eigenvalues and rank . Then contains a principal submatrix of order whose rank is equal to the rank of .
Proof. Let be the characteristic polynomial of . Since has nonzero eigenvalues, we have , and contains a principal nonzero minor. Using permutation similarity we may assume that the leading principal minor of , call it , is not equal to zero. Using another permutation similarity we may assume that the first rows of have rank , i.e. are linearly independent:
where is invertible, and has full rank . This implies that the matrix contains linearly independent columns. Using a permutation similarity that leaves the top left submatrix fixed, we can assure that the submatrix of containing the first rows and the first columns contains linearly independent columns. This implies that the top left submatrix of has rank .
The bound given in the above lemma cannot be improved in general as illustrated in the following example.
Let be an diagonal matrix with nonzero diagonal elements. The matrix
has order , rank and nonzero eigenvalues. It is easy to check that has no proper principal submatrices with rank .
Let be a nonnegative matrix with nonzero eigenvalues and rank . Then there exists a nonnegative matrix of order , whose nonzero spectrum is the same as the nonzero spectrum of and whose Jordan canonical form satisfies: .
Let be a diagonalizable nonnegative matrix of rank and . Then there exists a diagonalizable nonnegative matrix , whose nonzero spectrum is the same as the nonzero spectrum of .
Proof. For a diagonalizable matrix the rank of is equal to the number of nonzero eigenvalues of .
3 The Nonnegative Inverse Elementary Divisor Problem
The nonnegative inverse elementary divisor problem (NIEDP) asks for a given realizable spectrum , what are the possible Jordan forms of realising matrices. Of course, if has no repeated entries, this problem reduces to the NIEP for . It is conjectured, that if is realizable, then it is realizable by a nonnegative nonderogatory matrix, but this appears to be still open.
Minc  proved that if is diagonalizably realizable by a positive matrix , then for every Jordan form with spectrum , is realizable by a positive matrix similar to . This result is conjectured to hold, if we relax the condition that is positive to the assumption that is nonnegative, but this also seems to be open at present.
Now consider the classic example
We ask the questions what is the minimal for which is realizable by a nonnegative matrix with a given Jordan canonical form associated with , where is a diagonal matrix, is the Jordan canonical form with the minimal polynomial of degree , and is nonderogatory. We will denote the minimal in each case by .
It is shown in  that is realizable by a diagonalizable nonnegative matrix only for i.e. . In this case, the condition for diagonalizable realizability and symmetric realizability coincide. Also, in the same paper it is shown that
is realizable for , while by McDonald-Neumann inequality, given in , is necessary for symmetric realizability.
On the other hand, is realizable for . This is shown in , where a nonderogatory matrix with spectrum , is provided.
Here we present the matrix
has eigenvalues , it is nonnegative, and has Jordan canonical form for . This shows that .
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