The numerical range of 3 × 3 matrices

“…More recently, tests were developed allowing to determine which type of the numerical range occurs for a given 3 × 3 matrix [7], with the flat portion boundary situation further treated in [10]. Note that the exact knowledge of the eigenvalues is crucial for these tests.…”

mentioning

confidence: 99%

“…Further, we establish formulas, equivalent to the ones derived in [7], to determine whether the numerical range of a given 3 × 3 real matrix is the convex hull of an ellipse and a point. These formulas are given completely in terms of trace invariants which can be computed directly from the entries of the original matrix.…”

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3-by-3 matrices with elliptical numerical range revisited

Rault

Sendova

Spitkovsky

2013

ELA

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Abstract. According to Kippenhahn's classification, numerical ranges W (A) of unitarily irreducible 3 × 3 matrices A come in three possible shapes, an elliptical disk being one of them. The known criterion for the ellipticity of W (A) consists of several equations, involving the eigenvalues of A. It is shown herein that the set of 3 × 3 matrices satisfying these conditions is nowhere dense, i.e., one of the necessary conditions can be violated by an arbitrarily small perturbation of the matrix, and therefore by an insufficiently good numerical approximation of the eigenvalues.Moreover, necessary and sufficient conditions for a real A to have an elliptical W (A) are derived, involving only the matrix coefficients and not requiring the knowledge of the eigenvalues. A particular case of real companion matrices is considered in detail.

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“…The four basic shapes include a polygon (for normal matrices), the convex hull of an ellipse and a point, an ovular shape, and a shape with one flat portion on its boundary. Concrete tests to determine the shape from the entries of the 3 × 3 matrix are in [9]. Work has also been done to generalize Kippenhahn's result to 4 × 4 matrices [2].…”

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confidence: 99%

Trace conditions for symmetry of the numerical range

Deaett¹,

Lafuente-Rodriguez²,

Marín³

et al. 2013

ELA

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Abstract. A subset S of the complex plane has n-fold symmetry about the origin (n-sato) if z ∈ S implies e 2π n z ∈ S. The 3 × 3 matrices A for which the numerical range W (A) has 3-sato have been characterized in two ways. First, W (A) has 3-sato if and only if the spectrum of A has 3-sato while tr(A 2 A * ) = 0. In addition, W (A) has 3-sato if and only if A is unitarily similar to an element of a certain family of generalized permutation matrices. Here it is shown that for an n × n matrix A, if a specific finite collection of traces of words in A and A * are all zero, then W (A) has n-sato. Further, this condition is shown to be necessary when n = 4. Meanwhile, an example is provided to show that the condition of being unitarily similar to a generalized permutation matrix does not extend in an obvious way.

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The numerical range of 3 × 3 matrices

Cited by 136 publications

References 5 publications

Исследование Числовой Области Значений Одной Операторной Матрицы

Исследование Числовой Области Значений Одной Операторной Матрицы

3-by-3 matrices with elliptical numerical range revisited

Trace conditions for symmetry of the numerical range

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