Circulants and critical points of polynomials

Kushel, Olga Y.; Tyaglov, Mikhail

doi:10.1016/j.jmaa.2016.03.005

Cited by 10 publications

(6 citation statements)

References 17 publications

(30 reference statements)

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“…We then study differentiators following the work of Pereira [29] and show that Johnson's conjecture holds for spectra realizable via complex Hadamard similarities. Along the way, we provide an alternate proof to a result of Malamud [25] and extend the results on circulant matrices by Kushel and Tyaglov [20]. Additionally, we use these results to provide a new proof of a classical theorem on the interlacing roots and critical points for polynomials.…”

Section: Introductionmentioning

confidence: 71%

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On the realizability of the critical points of a realizable list

Hoover

McCormick

Paparella

et al. 2018

Linear Algebra and its Applications

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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.

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Section: Introductionmentioning

confidence: 71%

“…Cheung and Ng [5], recognize that other constructions of a d-companion matrix exist. They provide methods for constructing such matrices which have already proven fruitful in the study of circulant matrices [20]. Future research on finding alternative constructions to the d-companion matrix may be critical to solving this conjecture.…”

Section: Discussionmentioning

confidence: 99%

On the realizability of the critical points of a realizable list

Hoover

McCormick

Paparella

et al. 2018

Linear Algebra and its Applications

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“…Then every pn ´1q ˆpn ´1q principal submatrix of C has characteristic polynomial 1 n p 1 pzq. We note that, while the original statement of this result given in [KT16] is in terms of the pn ´1q pn ´1q upper left principal submatrix of C, the result and proof work for all pn ´1q ˆpn ´1q principal submatrices of C. In fact, any two pn ´1q ˆpn ´1q principal submatrices of C can be seen to be permutationally similar and so have the same characteristic polynomial.…”

Section: Spectra Of K-factor Positive Matricesmentioning

confidence: 91%

Absolutely $k$-Incoherent Quantum States and Spectral Inequalities for Factor Width of a Matrix

Johnston¹,

Moein²,

Pereira³

et al. 2022

Preprint

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We investigate the set of quantum states that can be shown to be k-incoherent based only on their eigenvalues (equivalently, we explore which Hermitian matrices can be shown to have small factor width based only on their eigenvalues). In analogy with the absolute separability problem in quantum resource theory, we call these states "absolutely k-incoherent", and we derive several necessary and sufficient conditions for membership in this set. We obtain many of our results by making use of recent results concerning hyperbolicity cones associated with elementary symmetric polynomials.

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“…( 1). From [16], the multiplication of IDFT and DFT with a circulant matrix yields a diagonal matrix to extract effective channel information.…”

Section: Introductionmentioning

confidence: 99%