Pfaffian and Decomposable Numerical Range of a Complex Skew Symmetric Matrix
نویسنده
چکیده
In the literature it is known that the decomposable numerical range W∧ k (A) of A ∈ Cn×n is not necessarily convex. But it is not known whether W∧ k (A) is star-shaped. We construct a symmetric unitary matrix A ∈ Cn×n such that the decomposable numerical range W∧ k (A) is not star-shaped and hence not simply connected. We then consider a real analog R∧ k (A) and show that R∧ k (A) is star-shaped if A ∈ Cn×n is skew symmetric. Such star-shapedness result is also true for the Pfaffian numerical range P∧ k (A).
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