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On the estimation of the \(q\)-numerical range of monic matrix polynomials. (English) Zbl 1065.15033

Summary: For a given \(q\in [0,1]\), the \(q\)-numerical range of an \(n\times n\) matrix polynomial \(P(\lambda)= I\lambda^m+ A_{m-1} \lambda^{m-1}+\cdots+ A_1\lambda+ A_0\) is defiend by \(W_q(P)= \{\lambda\in \mathbb C: y^*P(\lambda)x= 0\), \(x,y\in \mathbb C^n\), \(x^*x= y^*y=1\), \(y^*x=q\}\). In this paper, an inclusion-exclusion methodology for the estimation of \(W_q(P)\) is proposed. Our approach is based on i) the discretization of a region \(\Omega\) that contains \(W_q(P)\), and ii) the construction of an open circular disk, which does not intersect \(W_q(P)\), centered at every grid point \(\mu\in \Omega\setminus W_q(P)\). For the cases \(q=1\) and \(0< q<1\), an important difference arises in one of the steps of the algorithm. Thus, these two cases are discussed separately.

MSC:

15A60 Norms of matrices, numerical range, applications of functional analysis to matrix theory
15A22 Matrix pencils
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