On the q-numerical range of matrices and matrix polynomials
نویسندگان
چکیده
The q-numerical range (0 ≤ q ≤ 1) of an n × n matrix polynomial P (λ) = Amλ m + · · ·+ A1λ + A0 is defined by Wq(P ) = {λ ∈ C : y∗P (λ)x = 0, x, y ∈ C, x∗x = y∗y = 1, y∗x = q}. In this paper, we investigate the boundary and the shape of Wq(P ), using the notion of local dimension. We also obtain that the q-numerical range of first order matrix polynomials is always simply connected. Moreover, the special cases of 2×2 matrices and matrix polynomials are considered. In particular, the boundary of the q-numerical range of a 2 × 2 matrix polynomial of degree m lies on an algebraic curve of degree at most 8m.
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