Nonnegative realization of spectra having negative real parts

Laffey, Thomas J.; Šmigoc, Helena

doi:10.1016/j.laa.2005.12.008

Cited by 47 publications

(36 citation statements)

References 14 publications

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“…, a m−1 (u, v)) and A(u, v) = circ (a(u, v)) . For 1 ≤ ℓ ≤ m, let matrix S ℓ , related to the circulant blocks A(u, v) defined in (12). For 0 ≤ k ≤ m − 1, let ℓ=0 be m n-by-n complex matrices.…”

Section: Corollary 21 [1]mentioning

confidence: 99%

Guo's index for some classes of matrices

Robbiano

2019

Linear Algebra and its Applications

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A permutative matrix is a square matrix such that every row is a permutation of the first row. A circulant matrix is a matrix where each row is a cyclic shift of the row above to the right. The Guo's index λ 0 of a realizable list is the minimum spectral radius such that the list (up to the initial spectral radius) together with λ 0 is realizable. The Guo's index of some permutative matrices is obtained. Our results are constructive. Some examples designed using MATLAB are given at the end of the paper.

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Section: Corollary 21 [1]mentioning

confidence: 99%

Guo's index for some classes of matrices

Robbiano

2019

Linear Algebra and its Applications

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“…The Nonnegative Inverse Eigenvalue Problem (NIEP) is a problem to determine necessary and sufficient conditions for a list of n complex numbers to be realizable by a nonnegative square matrix A of order n. If the list Σ is realizable by a nonnegative matrix A, then we say that A realizes Σ or it is a realizing matrix for Σ. Some results can be seen in [9,10,11]. This very difficult problem attracted the attention of many authors over the last 50 years, and it was firstly considered by Suleȋmanova in 1949 (see [26]).…”

Section: Introductionmentioning

confidence: 99%

On the spectra of some g-circulant matrices and applications to nonnegative inverse eigenvalue problem

Andrade

Arrieta

Manzaneda

et al. 2020

Linear Algebra and its Applications

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A g-circulant matrix of order n is defined as a matrix of order n where each row is a right cyclic shift in g−places to the preceding row. Using number theory, certain nonnegative g-circulant real matrices are constructed. In particular, it is shown that spectra with sufficient conditions so that it can be the spectrum of a real g-circulant matrix is not a spectrum with sufficient conditions so that it can be the spectrum of a real circulant matrix of the same order. The obtained results are applied to Nonnegative Inverse Eigenvalue Problem to construct nonnegative, g-circulant matrices with given appropriated spectrum. Moreover, nonnegative g-circulant matrices by blocks are also studied and in this case, their orders can be a multiple of a prime number.

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“…However, the NIEP in which we may augment the list σ by adding an arbitrary number N of zeros was solved theoretically by Boyle and Handelman [1] and a constructive version was found by La ey [14]. For special cases that bound the size of N we refer the reader to [3], [18], [19] The rst signi cant result on NIEP was Suleimanova's result [32] on lists of real numbers with just one positive number, which says that the real list λ > ≥ λ ≥ · · · ≥ λn is realizable if and only if λ + λ +· · ·+ λn ≥ . The question of realizing real lists of ve or more numbers containing just two positive numbers is still unsolved in general ( [13], [2]).…”

Section: Introductionmentioning

confidence: 99%

The Diagonalizable Nonnegative Inverse Eigenvalue Problem

Cronin

Laffey

2018

Special Matrices

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Abstract:In this article we provide some lists of real numbers which can be realized as the spectra of nonnegative diagonalizable matrices but which are not the spectra of nonnegative symmetric matrices. In particular, we examine the classical list σ = ( + t, − t, − , − , − ) with t ≥ , and show that σ is realizable by a nonnegative diagonalizable matrix only for t ≥ . We also provide examples of lists which are realizable as the spectra of nonnegative matrices, but not as the spectra of nonnegative diagonalizable matrices by examining the Jordan Normal Form.

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Nonnegative realization of spectra having negative real parts

Cited by 47 publications

References 14 publications

Guo's index for some classes of matrices

Guo's index for some classes of matrices

On the spectra of some g-circulant matrices and applications to nonnegative inverse eigenvalue problem

The Diagonalizable Nonnegative Inverse Eigenvalue Problem

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