Applications of a Brauer theorem in the nonnegative inverse eigenvalue problem

Soto, Ricardo L.; Rojo, Óscar

doi:10.1016/j.laa.2005.12.026

Cited by 64 publications

(57 citation statements)

References 29 publications

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“…This result was first used by Perfect [14] in connection with the NIEP, and has given rise lately to a number of realizability criteria [19,20,22]. Closer to our approach in this paper is Rado's 1 extension (Theorem 2.3 below) of Brauer's result, which was used by 2.…”

Section: Introductionmentioning

confidence: 92%

“…One of these cases is shown to be not realizable as the spectrum of a symmetric nonnegative matrix by using a necessary condition given in ( [13], Lemma 4.1). Concerning to the second unresolved case, it is shown in [22] that every point on the line from l to m is also realized by a symmetric nonnegative circulant matrix. Conditions…”

Section: Comparison With Previous Criteriamentioning

confidence: 99%

“…The complex cases n = 4 and n = 5 have been solved for matrices of trace zero by Reams [17] and Laffey and Meehan [10], respectively. Sufficient conditions or realizability criteria for the existence of a nonnegative matrix with a given real spectrum have been obtained in [25,14,15,18,8,1,19,22] (see [3, §2.1] and references therein for a comprehensive survey). If we additionally require the realizing matrix to be symmetric, we have the symmetric nonnegative inverse eigenvalue problem (hereafter SNIEP).…”

Section: Introductionmentioning

confidence: 99%

“…Perfect used this extension to derive a realizability criterion for the RNIEP. Although it turns out to be a quite powerful result, inexplicably, this criterion was completely ignored for many years in the literature until it was brought up again in [22]. Perfect's criterion for the RNIEP is the following: …”

mentioning

confidence: 99%

“…(see also [22]) where To show the power of both Theorem 2.3 and Theorem 2.4, consider the following example; in which, as far as we know, no other realizability criterion is satisfied by the set Λ (except the extended Perfect criterion given in [22]): Example 2.5. Let Λ = {6, 3, 3, −5, −5}.…”

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confidence: 99%

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Symmetric nonnegative realization of spectra

Soto¹,

Rojo²,

Moro³

et al. 2007

ELA

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Abstract.A perturbation result, due to R. Rado and presented by H. Perfect in 1955, shows how to modify r eigenvalues of a matrix of order n, r ≤ n, via a perturbation of rank r, without changing any of the n − r remaining eigenvalues. This result extended a previous one, due to Brauer, on perturbations of rank r = 1. Both results have been exploited in connection with the nonnegative inverse eigenvalue problem. In this paper a symmetric version of Rado's extension is given, which allows us to obtain a new, more general, sufficient condition for the existence of symmetric nonnegative matrices with prescribed spectrum.Key words. Symmetric nonnegative inverse eigenvalue problem.

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Section: Introductionmentioning

confidence: 92%

Section: Comparison With Previous Criteriamentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

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Symmetric nonnegative realization of spectra

Soto¹,

Rojo²,

Moro³

et al. 2007

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A family of realizability criteria for the real and symmetric nonnegative inverse eigenvalue problem

Soto

2011

Numerical Linear Algebra App

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A new realizability criterion for the real nonnegative inverse eigenvalue problem is introduced. This criterion is a nontrivial extension of a powerful previous sufficient condition, based on negativity compensation. If the criterion is satisfied, then we can always construct a realizing matrix. It is also proved that this new criterion is easily adaptable to be sufficient for the construction of a symmetric nonnegative matrix with given spectrum. In a natural way, the criterion extends to a family of sufficient conditions for the problem to have a solution.

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Generalization of the Brauer Theorem to Matrix Polynomials and Matrix Laurent Series

Бини

Meini

2017

Large Truncated Toeplitz Matrices, Toeplitz Operators, and Related Topics

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Given a square matrix A, Brauer's theorem [Duke Math. J. 19 (1952), [75][76][77][78][79][80][81][82][83][84][85][86][87][88][89][90][91] shows how to modify one single eigenvalue of A via a rank-one perturbation, without changing any of the remaining eigenvalues. We reformulate Brauer's theorem in functional form and provide extensions to matrix polynomials and to matrix Laurent series A(z) together with generalizations to shifting a set of eigenvalues. We provide conditions under which the modified function A(z) has a canonical factorization A(z) = U (z) L(z −1 ) and we provide explicit expressions of the factors U (z) and L(z). Similar conditions and expressions are given for the factorization of A(z −1 ). Some applications are discussed. * Work supported by Gruppo Nazionale di Calcolo Scientifico (GNCS) of INdAM, and by the PRA project "Mathematical models and computational methods for complex networks" funded by the University of Pisa. inverse eigenvalue problem [10], [21], to accelerating the convergence of iterative methods for solving matrix equations in particular quadratic equations [20], equations related to QBD Markov chains [17], [14], to M/G/1-type Markov chains [5], [3], or for algebraic Riccati equations in [15], [16], [19], and [2].In this paper we revisit Brauer's theorem in functional form and generalize it to a non-degenerate n × n matrix function A(z) analytic in the open annulusHere, non-degenerate means that det A(z) is not identically zero. Throughout the paper we assume that all the matrix functions are nondegenerate.Let λ ∈ A r1,r2 be an eigenvalue of A(z), i.e., such that det A(λ) = 0 and u ∈ C n , u = 0, be a corresponding eigenvector, i.e., such that A(λ)u = 0. We prove that for any v ∈ C n such that v * u = 1 and for any µ ∈ C, the functionwhere v * denotes the transpose conjugate of v. Moreover, A(z) has all the eigenvalues of A(z), except for λ which, if µ ∈ A r1,r2 , is replaced by µ, otherwise is removed. We provide explicit expressions which relate the coefficients A i of A(z) = i∈Z z i A i to those of A(z).This transformation, called right shift, is obtained by using a right eigenvector u of A(z). A similar version, called left shift, is given by using a left eigenvector v of A(z). Similarly, a combination of the left and the right shift, leads to a transformation called double shift, which enables one to shift a pair of eigenvalues by operating to the right and to the left of A(z).This result, restricted to a matrix Laurent polynomial A(z) = k −h z i A i , provides a shifted function which is still a matrix Laurent polynomial A(z) = k i=−h z i A i . In the particular case where A(z) = zI − A, we find that A(z) = zI − A is such that A is the matrix given in the classical theorem of Brauer [9]. Therefore, these results provide an extension of Brauer's Theorem 1.1 to matrix Laurent series and to matrix polynomials.A further generalization is obtained by performing the right, left, or double shift simultaneously to a packet of eigenvalues once we are given an invariant (right or le...

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Applications of a Brauer theorem in the nonnegative inverse eigenvalue problem

Cited by 64 publications

References 29 publications

Symmetric nonnegative realization of spectra

Symmetric nonnegative realization of spectra

A family of realizability criteria for the real and symmetric nonnegative inverse eigenvalue problem

Generalization of the Brauer Theorem to Matrix Polynomials and Matrix Laurent Series

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