On the numerical range map

Joswig, Michael; Straub, Bernd F.

doi:10.1017/s1446788700034996

Cited by 19 publications

(27 citation statements)

References 16 publications

(41 reference statements)

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“…The algebraic and geometric properties of the map Φ A have been extensively studied, see [19,26,4,9,16,17]. In particular, the set of all critical values of Φ A , which we denote by Σ A ⊂ C, has received a lot of attention, because this is an interesting object which contains a lot of information on the matrix A.…”

Section: Introductionmentioning

confidence: 99%

Numerical Measure of a Complex Matrix

Gallay

Serre

2011

Comm Pure Appl Math

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International audienceWe introduce a natural probability measure over the numerical range of a complex matrix A ∈ M n (C). This numerical measure µ A can be defined as the law of the random variable X ∈ C when the vector X ∈ C n is uniformly distributed on the unit sphere. If the matrix A is normal, we show that µ A has a piecewise polynomial density f A , which can be identified with a multivariate B-spline. In the general (nonnormal) case, we relate the Radon transform of µ A to the spectrum of a family of Hermitian matrices, and we deduce an explicit representation formula for the numerical density which is appropriate for theoretical and computational purposes. As an application, we show that the density f A is polynomial in some regions of the complex plane which can be characterized geometrically, and we recover some known results about lacunas of symmetric hyperbolic systems in 2 + 1 dimensions. Finally, we prove under general assumptions that the numerical measure of a matrix A ∈ M n (C) concentrates to a Dirac mass as the size n goes to infinity

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Section: Introductionmentioning

confidence: 99%

Numerical Measure of a Complex Matrix

Gallay

Serre

2011

Comm Pure Appl Math

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“…We start with some facts concerning the algebraic curve C A the convex hull of which is F (A). This curve coincides with critical values of the map (1.1) when the domain CS n and the target space C are treated as real manifolds [8]. Let Σ A denote the set of all critical values of f A , then the curve C A ⊆ Σ A .…”

Section: Inverse Continuitymentioning

confidence: 74%

“…Let Σ A denote the set of all critical values of f A , then the curve C A ⊆ Σ A . In fact, if we include the bitangent set C A containing all line segments connecting two points in C A when the points have the same tangent, then Σ A = C A ∪ C A [8]. The following description of the sets C A and Σ A is based on [10] and [8]; see [5,Section 5] for more details.…”

Section: Inverse Continuitymentioning

confidence: 99%

Inverse continuity on the boundary of the numerical range

Leake

Lins

Spitkovsky

2013

Linear and Multilinear Algebra

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Abstract. Let A ∈ Mn(C). We consider the mapping f A (x) = x * Ax, defined on the unit sphere in C n . The map has a multi-valued inverse f A are considered in terms of the structure of the set of pre-images for points in the numerical range. It is shown that there may be only finitely many failures of continuity of f −1 A , and conditions for where these failure occur are given. Additionally, we give a necessary and sufficient condition for weak inverse continuity to hold for n = 4 and a sufficient condition for n > 4.

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“…We recall that the derivative of λ k with respect to θ (see Lemma 3.2 of [19] and Sec. 5 of [11]) is…”

Section: Representation Of Extreme Pointsmentioning

confidence: 99%

Pre-images of extreme points of the numerical range, and applications

Spitkovsky¹,

Weis

2016

Operators and Matrices

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Abstract. We extend the pre-image representation of exposed points of the numerical range of a matrix to all extreme points. With that we characterize extreme points which are multiply generated, having at least two linearly independent pre-images, as the extreme points which are Hausdorff limits of flat boundary portions on numerical ranges of a sequence converging to the given matrix. These studies address the inverse numerical range map and the maximum-entropy inference map which are continuous functions on the numerical range except possibly at certain multiply generated extreme points. This work also allows us to describe closures of subsets of 3-by-3 matrices having the same shape of the numerical range.Mathematics subject classification (2010): 47A12, 54C10, 62F30, 94A17.

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On the numerical range map

Cited by 19 publications

References 16 publications

Numerical Measure of a Complex Matrix

Numerical Measure of a Complex Matrix

Inverse continuity on the boundary of the numerical range

Pre-images of extreme points of the numerical range, and applications

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