[Paper Review] Diagonal elements in the Nonnegative Inverse Eigenvalue Problem

Richard Ellard, Helena Šmigoc|arXiv (Cornell University)|Feb 8, 2017

Advanced Combinatorial Mathematics5 references3 citations

TL;DR

This paper solves a key aspect of the Nonnegative Inverse Eigenvalue Problem (NIEP) by characterizing which nonnegative diagonal entries can appear in a nonnegative matrix with a given spectrum, where one eigenvalue is nonnegative and all others have nonpositive real parts. It proves that such a diagonal list is realizable if and only if the sum of the diagonal entries matches the trace and the sum of their squares does not exceed the sum of squares of the eigenvalues, with realization achieved via a companion matrix plus diagonal matrix structure.

ABSTRACT

We say that a list of complex numbers is "realisable" if it is the spectrum of some (entrywise) nonnegative matrix. The Nonnegative Inverse Eigenvalue Problem (NIEP) is the problem of characterising all realisable lists. Although the NIEP remains unsolved, it has been solved in the case where every entry in the list (apart from the Perron eigenvalue) has nonpositive real part. For a given spectrum of this type, we show that a list of nonnegative numbers may arise as the diagonal elements of the realising matrix if and only if these numbers satisfy a remarkably simple inequality. Furthermore, we show that realisation can be achieved by the sum of a companion matrix and a diagonal matrix.

Motivation & Objective

• To determine the necessary and sufficient conditions for a list of nonnegative numbers to appear as diagonal entries of a nonnegative matrix with a given spectrum.
• To extend known results on the NIEP—particularly the Laffey-Šmigoc theorem—by incorporating constraints on diagonal elements.
• To show that realizations can be constructed as the sum of a companion matrix and a diagonal matrix, preserving spectral and diagonal constraints.
• To characterize the range of possible diagonal elements (minimum and maximum) for such realizable spectra.

Proposed method

• Uses power sums $ s_1 $ and $ s_2 $ of the spectrum and diagonal entries to derive necessary and sufficient conditions for realizability.
• Applies Newton's identities to relate the coefficients of the characteristic polynomial to the diagonal and subdiagonal entries of a structured matrix.
• Constructs a matrix of the form $ A = C + D $, where $ C $ is a companion matrix with zero trace and $ D $ is diagonal, to realize the desired spectrum and diagonal entries.
• Employs the Cauchy-Schwarz inequality to bound deviations of diagonal entries from the mean, linking $ s_2 $-norms to spectral constraints.
• Derives explicit formulas for subdiagonal entries $ b_k $ in terms of power sums and symmetric functions, ensuring nonnegativity.
• Validates realizability by showing that the derived $ b_k $ values are nonnegative under the stated inequalities.

Experimental results

Research questions

• RQ1 What conditions must a list of nonnegative diagonal entries satisfy to be realizable as the diagonal of a nonnegative matrix with a given spectrum where one eigenvalue is nonnegative and all others have nonpositive real parts?
• RQ2 Can the realization of such spectra be achieved using a matrix structure that is the sum of a companion matrix and a diagonal matrix?
• RQ3 What is the range of possible values for individual diagonal entries in such realizations?
• RQ4 How do the power sums $ s_1 $ and $ s_2 $ of the diagonal entries relate to those of the spectrum in a realizable case?
• RQ5 Is the ordering of diagonal entries in the matrix structure critical for realizability, and if so, why?

Key findings

• A list $ ho, ho_2, ho_3, ho_4 $ with $ ho eq 0 $ and $ ext{Re}( ho_i) eq 0 $ for $ i eq 1 $ is realizable as a nonnegative matrix if and only if $ s_1( ext{diag}) = s_1( ext{spec}) $ and $ s_2( ext{diag}) eq s_2( ext{spec}) $.
• The diagonal entries $ a_1, a_2, ext{...}, a_n $ must satisfy $ s_1( ext{diag}) = s_1( ext{spec}) $ and $ s_2( ext{diag}) eq s_2( ext{spec}) $, which is both necessary and sufficient for realizability.
• Realization is possible via a matrix of the form $ A = C + D $, where $ C $ is a companion matrix with zero trace and $ D $ is diagonal, ensuring nonnegativity of entries.
• The minimal and maximal possible diagonal entries for a given spectrum are bounded by $ 0 eq a eq s_1( ext{spec}) $ and $ (a - s_1/n)^2 eq (n-1)(ns_2 - s_1^2)/n^2 $, respectively.
• The ordering of diagonal entries in the matrix structure is essential: if $ a_1 eq a_2 eq ext{...} eq a_n $ is not non-increasing, the resulting matrix may not be nonnegative, as shown in Example 5.5.
• When the JLL condition $ s_1^2 eq n s_2 $ holds with equality, the only possible realization has constant diagonal entries.