On the realizability of the critical points of a realizable list

Abstract

The nonnegative inverse eigenvalue problem (NIEP) is to characterize the spectra of entrywise nonnegative matrices. A finite multiset of complex numbers is called realizable if it is the spectrum of an entrywise nonnegative matrix. Monov conjectured that the k th -moments of the list of critical points of a realizable list are nonnegative. Johnson further conjectured that the list of critical points must be realizable. In this work, Johnson's conjecture, and consequently Monov's conjecture, is established for a variety of important cases including Ciarlet spectra, Suleȋmanova spectra, spectra realizable via companion matrices, and spectra realizable via similarity by a complex Hadamard matrix. Additionally we prove a result on differentiators and trace vectors, and use it to provide an alternate proof of a result due to Malamud and a generalization of a result due to Kushel and Tyaglov on circulant matrices. Implications for further research are discussed.

Integrability of matrices

“…Let now K be an arbitrary algebraically closed field of characteristic zero. As it was mentioned above, system (7) for a Shabat polynomial reduces to a systems of equations in a 1 , . .…”

“…After fixing critical values of a Shabat polynomial, we still have a "degree of freedom" corresponding to a choice of µ 2 . Thus, we can impose some further restrictions on system (7). For example, we can assume that a 1 = 0 and b 1 = 1.…”

“…Theorem 2.6 implies that in this case the system (7) has only finitely many solutions. Since (7) provide us with equations in a 1 , . .…”

“…The converse operation, namely, the operation of integration, is due to Bhat and Mukherjee ( [1]). Notice that the inegrability of matrices has applications to the nonnegative inverse eigenvalue problem and to inequalities like the dual Schoenberg type inequality (see [7], [5] and references therein).…”