Indefinite numerical range of 3 × 3 matrices

Bebiano, Natália; Providência, João da; Teixeira, Ricardo Emanuel Cunha

doi:10.1007/s10587-009-0016-4

Cited by 3 publications

(7 citation statements)

References 16 publications

(20 reference statements)

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“…We recall that an usual procedure to find the point equation of the boundary generating curve C (A, B) is to eliminate one of the indeterminates, say u, from (3) and ux + vy + w = 0, dehomogenize the result by setting w = 1, and to eliminate the remaining parameter v from the equations…”

Section: The Polynomial F (U V W) and W (A B)mentioning

confidence: 99%

“…The corresponding result for B indefinite is as follows [3]. For details on plane algebraic curves, we refer the interested reader to [5].…”

Section: The Polynomial F (U V W) and W (A B)mentioning

confidence: 99%

“…For arbitrary A ∈ M 3 , W (A, B) has been characterized in [3], following Kippenhahn's approach in the classical case.…”

Section: C(a B) For B Indefinite and A Arbitrarymentioning

confidence: 99%

“…There are examples showing that all the above curves may occur as boundary generating curves [3]. The characterization of W (A, B) requires the determination of the signs of each branch of C(A, B), in order to obtain the pseudo-convex hull of the boundary generating curve.…”

Section: Theorem 8 If the Pencil (A B) Is In N D Then The Pseudo-cmentioning

confidence: 99%

“…If the matrix B is nonsingular, the spectrum of the regular pencil denoted by σ (A, B) consists of all the zeros λ of the polynomial det(A − λ B). The spectral theory of pencils is an important issue in pure mathematics as well as in applications (e.g., see [3,8,12,13] and their references). An informative containment region for the spectrum of (A, B) is the numerical range.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

The Characteristic Polynomial of Linear Pencils of Small Size and the Numerical Range

Bebiano

Providência

Esmaeili

2017

Springer Proceedings in Mathematics &Amp; Statistics

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The numerical range of a linear pencil (A, B) of matrices of size n may be characterized in terms of a certain algebraic curve of class n, called the boundary generating curve, which is explicitly given by the characteristic polynomial of the pencil. We shall be specially concerned with the case of one of the matrices being Hermitian. For n = 2 and n = 3, each possible type of boundary generating curve can be completely described. For n = 3, the curve type is given by Newton's classification of cubic curves. Illustrative examples of the different possibilities are given.

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Section: The Polynomial F (U V W) and W (A B)mentioning

confidence: 99%

“…The corresponding result for B indefinite is as follows [3]. For details on plane algebraic curves, we refer the interested reader to [5].…”

Section: The Polynomial F (U V W) and W (A B)mentioning

confidence: 99%

“…For arbitrary A ∈ M 3 , W (A, B) has been characterized in [3], following Kippenhahn's approach in the classical case.…”

Section: C(a B) For B Indefinite and A Arbitrarymentioning

confidence: 99%

Section: Theorem 8 If the Pencil (A B) Is In N D Then The Pseudo-cmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

The Characteristic Polynomial of Linear Pencils of Small Size and the Numerical Range

Bebiano

Providência

Esmaeili

2017

Springer Proceedings in Mathematics &Amp; Statistics

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Ratio Numerical Ranges of Operators

Rodman

Spitkovsky

2011

Integr. Equ. Oper. Theory

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Geometric properties of ratio numerical ranges of two Hilbert space operators are studied. Characterizations in terms of algebraic properties of operators are given for connectedness of ratio numerical ranges, and for the situations when the ratio numerical ranges are contained in a circle or in a line. Under a hypothesis of sectoriality (in a weak sense) simple connectedness of ratio numerical ranges is proved, and their boundedness property characterized.Mathematics Subject Classification (2010). Primary 47A12.

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Computing the numerical range of Krein space operators

et al. 2014

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Consider the Hilbert space (H; h ; i) equipped with the indefinite inner product OEu; v D v J u, u; v 2 H, where J is an indefinite self-adjoint involution acting on H. The Krein space numerical range W J .T / of an operator T acting on H is the set of all the values attained by the quadratic form OET u; u, with u 2 H satisfying OEu; u D˙1. We develop, implement and test an alternative algorithm to compute W J .T / in the finite dimensional case, constructing 2 by 2 matrix compressions of T and their easily determined elliptical and hyperbolical numerical ranges. The numerical results reported here indicate that this method is very efficient, since it is faster and more accurate than either of the existing algorithms. Further, it may yield easy solutions for the inverse indefinite numerical range problem. Our algorithm uses an idea of Marcus and Pesce from 1987 for generating Hilbert space numerical ranges of matrices of size n.

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Indefinite numerical range of 3 × 3 matrices

Cited by 3 publications

References 16 publications

The Characteristic Polynomial of Linear Pencils of Small Size and the Numerical Range

The Characteristic Polynomial of Linear Pencils of Small Size and the Numerical Range

Ratio Numerical Ranges of Operators

Computing the numerical range of Krein space operators

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