The validity of the Marcus-de Oliveira conjecture for essentially Hermitian matrices

Bebiano, Natália; Kovačec, Alexander; Providência, João da

doi:10.1016/0024-3795(94)90498-7

Cited by 12 publications

(5 citation statements)

References 9 publications

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“…We are interested in obtaining determinantal inequalities for J-accretive dissipative matrices. Determinantal inequalities have deserved the attention of researchers, [2], [3], [5]- [9], [11].…”

Section: Resultsmentioning

confidence: 99%

Determinantal inequalities for J-accretive dissipative matrices

Bebiano¹,

Providência²

2017

Stud. Univ. Babes-Bolyai Math.

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Section: Resultsmentioning

confidence: 99%

Determinantal inequalities for J-accretive dissipative matrices

Bebiano¹,

Providência²

2017

Stud. Univ. Babes-Bolyai Math.

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“…In Example 3.5 the eigenvalues lie on a circle centered at the origin. In Example 3.6 the eigenvalues lie on a circle centered at 5 2 − 3 2 i.…”

Section: Thenmentioning

confidence: 99%

“…It is clear that A (C) = (−1) n C (A) and this set is unitarily invariant, that is, conjecture was proved for n ≤ 3 and for particular classes of matrices (see [2,3,5,10] and references therein).…”

Section: Introductionmentioning

confidence: 96%

On the C-determinantal range for special classes of matrices

Guterman

Lemos

Soares

2016

Applied Mathematics and Computation

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Abstract. Let A and C be square complex matrices of size n, the C-determinantal range of A is the subset of the complex plane {det (A − U CU * ) : U U * = I n }. If A, C are both Hermitian matrices, then by a result of M. Fiedler [11] this set is a real line segment.In this paper we study this set for the case when C is a Hermitian matrix. Our purpose is to revisit and improve two well-known results on this topic. The first result is due to C.-K. Li concerning the C-numerical range of a Hermitian matrix, see Condition 5.1 (a) in [20]. The second one is due to C.-K. Li, Y.-T. Poon and N.-S. Sze about necessary and sufficient conditions for the C-determinantal range of A to be a subset of the line, see [21, Theorem 3.3].

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“…If A is θ-Hermitian then A is normal because θA is Hermitian, and by the spectral theorem for Hermitian matrices, all eigenvalues of A lie on the line : R → C : ρ → ρθ. In the literature, for instance in [1,2,5], A is called essentially Hermitian if there exists an α ∈ C such that A − αI is θ-Hermitian for some θ ∈ T. Clearly, the spectrum of an essentially Hermitian matrix lies on an affine line shifted over α ∈ C. Conversely, if a normal matrix has all its eigenvalues on a line ⊂ C, it is essentially Hermitian. This includes all normal 2 × 2 matrices and all normal rank one perturbations of αI for α ∈ C. Larger and higher rank normal matrices have their eigenvalues on a polynomial curve C ⊂ C of higher degree.…”

Section: Normal Matrices With All Eigenvalues On Polynomial Curvesmentioning

confidence: 99%

“…In this paper, we study the perturbation of normal matrices by essentially Hermitian matrices [1,2,5]. These are matrices E that can be written as E = βH + αI, where α, β ∈ C, H is Hermitian, and I is the identity matrix.…”

Section: Introductionmentioning

confidence: 99%

Normalitity preserving perturbations and augmentations and their effect on the eigenvalues

da Silva,

Brandts

2011

Preprint

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We revisit the normality preserving augmentation of normal matrices studied by Ikramov and Elsner in 1998 and complement their results by showing how the eigenvalues of the original matrix are perturbed by the augmentation. Moreover, we construct all augmentations that result in normal matrices with eigenvalues on a quadratic curve in the complex plane, using the stratification of normal matrices presented by Huhtanen in 2001. To make this construction feasible, but also for its own sake, we study normality preserving normal perturbations of normal matrices. For 2 × 2 and for rank-1 matrices, the analysis is complete. For higher rank, all essentially Hermitian normality perturbations are described. In all cases, the effect of the perturbation on the eigenvalues of the original matrix is given. The paper is concluded with a number of explicit examples that illustrate the results and constructions.

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The validity of the Marcus-de Oliveira conjecture for essentially Hermitian matrices

Cited by 12 publications

References 9 publications

Determinantal inequalities for J-accretive dissipative matrices

Determinantal inequalities for J-accretive dissipative matrices

On the C-determinantal range for special classes of matrices

Normalitity preserving perturbations and augmentations and their effect on the eigenvalues

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