## 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

Zbl 0899.15017

Mathematica
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