Flat portions on the boundary of the Davis–Wielandt shell of 3-by-3 matrices

Chien, Mao-Ting; Nakazato, Hiroshi

doi:10.1016/j.laa.2008.07.014

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2018

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“…An analysis of singularities of S F for d = n = 3 shows that W (F ) has at most four large faces which are ellipses [14]. This estimate also follows from our classification.…”

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confidence: 75%

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Classification of joint numerical ranges of three hermitian matrices of size three

Szymański

Weis

Życzkowski

2018

Linear Algebra and its Applications

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Abstract. The joint numerical range W (F ) of three hermitian 3-by-3 matrices F = (F 1 , F 2 , F 3 ) is a convex and compact subset in R 3 . We show that W (F ) is generically a three-dimensional oval. Assuming dim(W (F )) = 3, every one-or two-dimensional face of W (F ) is a segment or a filled ellipse. We prove that only ten configurations of these segments and ellipses are possible. We identify a triple F for each class and illustrate W (F ) using random matrices and dual varieties.

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“…An analysis of singularities of S F for d = n = 3 shows that W (F ) has at most four large faces which are ellipses [14]. This estimate also follows from our classification.…”

supporting

confidence: 75%

“…Despite the long history of the problem [8,37,14,15,12], a classification of the joint numerical range of triples of matrices (n = 3) is unknown even in the case d = n = 3. Our motivation to tackle this problem is quantum mechanics, as we explain in Section 2.…”

Section: Introductionmentioning

confidence: 99%