Original scientific paper

Cubic surfaces and q-numerical ranges

Mao-Ting Chien ; Department of Mathematics, Soochow University, Taiwan
Hiroshi Nakazato ; Department of Mathematical Sciences, Faculty of Science and Technology, Hirosaki University, Japan

Abstract

Let $A$ be an $n \times n$ complex matrix and $0\leq q \leq 1$. The boundary of the $q$-numerical range of $A$ is the orthogonal projection of a hypersurface defined by the dual surface of the homogeneous polynomial
\[F(t, x, y, z)= {\rm det}(t\, I_n +x(A +A^*)/2+ y(A -A^*)/(2i) +z \, A^* A).
\]
We construct different types of cubic surfaces $S_F$ corresponding to the homogeneous polynomial $F(t, x, y, z)$ induced by some $3\times 3$ matrices. The degree of the boundary of the Davis-Wielandt shell of a $3 \times 3$ upper triangular matrix is determined by the cubic surface~$S_F$.

Keywords

singular points; cubic surfaces; q-numerical range; Davis-Wielandt shell

Hrčak ID:

101406

URI

https://hrcak.srce.hr/101406

Publication date:

10.5.2013.

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