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The boundary of the Q-numerical range of some Toeplitz nilpotent matrix. (English) Zbl 1383.15020

Let \(A\) be an \(n\times n\) complex matrix. The usual numerical range \(W(A)\) is defined to be set of all \(\xi^{\ast}A\xi\) (\(\xi\in\mathbb{C}^{n}\), \(\left\| \xi\right\| =1\)). For \(\left| q\right| \leq1\) the \(q\)-numerical range \(W_{q}(A)\) consists of all \(\eta^{\ast}A\xi\) (\(\eta,\xi\in\mathbb{C} ^{n}\), \(\left\| \eta\right\| =\left\| \xi\right\| =1\) and \(\eta^{\ast}\xi=q\)) (see, for example [C.-K. Li and H. Nakazato, Linear Multilinear Algebra 43, No. 4, 385–409 (1998; Zbl 0899.15017)]). The authors are interested in computing the boundary of \(W_{q}(A)\). This has been done by other authors for some typical \(3\times3\) matrices and it is believed that the boundary for a generic \(3\times3\) matrix can be described by a polynomial of degree \(24\). In the present paper the authors consider a particular \(4\times4\) nilpotent matrix \(N\) with rows \(0~1~0~1;~0~0~1~0;~0~0~0~1;~0~0~0~0\). Taking \(q=1599/1601\) (part of a Pythagorean triple so that the polynomial is rational) they construct a homogeneous polynomial \(g(x,y)\) with \(253\) terms and of degree \(40\) such that the boundary points \(x+iy\) of \(W_{q}(N)\) are the roots of \(g(x,y)\). Similarly they construct a homogeneous rational polynomial \(L_{0,N}(X,Y,Z)\) with \(135\) terms and of degree \(12\) whose roots are determined by the boundary points \(X+iY=\xi^{\ast}N\xi\) of \(W(N)\) and \(Z=\xi^{\ast}N^{\ast}N\xi\). Some Mathematica code to compute the boundary points numerically is included.

MSC:

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

Citations:

Zbl 0899.15017

Software:

Mathematica
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Full Text: Euclid