Topic
Complex polytope
About: Complex polytope is a research topic. Over the lifetime, 11 publications have been published within this topic receiving 206 citations. The topic is also known as: complex polytopes.
Papers
TL;DR: The small complex polytope extremality theorem is proved under some more restrictive hypotheses on the underlying family of matrices, which assures a certain finiteness property on the number of vertices of the unit ball of the extremal complexpolytope norm.
Abstract: In this paper we consider finite families of complex $n\times n$-matrices. In particular, we focus on those families that satisfy the so-called finiteness conjecture, which was recently disproved in its more general formulation. We conjecture that the validity of the finiteness conjecture for a finite family of nondefective type is equivalent to the existence of an extremal norm in the class of complex polytope norms. However, we have not been able to prove this complex polytope extremality conjecture, but we are able to prove the small complex polytope extremality theorem under some more restrictive hypotheses on the underlying family of matrices. In addition, our theorem assures a certain finiteness property on the number of vertices of the unit ball of the extremal complex polytope norm, which could be very useful for the construction of suitable algorithms aimed at the actual computation of the spectral radius of the family.
97 citations
TL;DR: This paper focuses on the computation of the joint spectral radius $\rho({\cal F})$ via the detection of an extremal norm in the class of complex polytope norms, whose unit balls are balanced complex polytopes with a finite essential system of vertices.
Abstract: In this paper we consider finite families ${\cal F}$ of real $n\!\times\!n$ matrices In particular, we focus on the computation of the joint spectral radius $\rho({\cal F})$ via the detection of an extremal norm in the class of complex polytope norms, whose unit balls are balanced complex polytopes with a finite essential system of vertices Such a finiteness property is very useful in view of the construction of efficient computational algorithms More precisely, we improve the results obtained in our previous paper [N Guglielmi, F Wirth, and M Zennaro, SIAM J Matrix Anal Appl, 27 (2005), pp 721-743], where we gave some conditions on the family ${\cal F}$ which are sufficient to guarantee the existence of an extremal complex polytope norm Unfortunately, they exclude unnecessarily many interesting cases of real families Therefore, here we relax the conditions given in our previous paper in order to provide a more satisfactory treatment of the real case
40 citations
1 Jan 2007
TL;DR: In this paper, the notion of balanced complex polytope was introduced as a generalization of real polytopes to the complex space C n, and the geometric properties of such complex polytopes and of their counterparts in the adjoint form were studied.
Abstract: In this paper we study the notion of balanced complex polytope as a generalization of a symmetric real polytope to the complex space C n . We pay particular attention to the geometric properties of such complex polytopes and of their counterparts in the adjoint form. In particular, we stress the differences occurring with respect to the well-known real case. We also introduce and discuss the related definitions of complex polytope norm and adjoint complex polytope norm.
22 citations
TL;DR: In this article, the authors define a finite class of tight frames of n vectors for C d (n and d fixed) which they call the highly symmetric tight frames and outline how these frames can be calculated from the representations of abstract groups using a computer algebra package.
20 citations
TL;DR: It is given two counterexamples that show that the conjecture that any nondefective set of matrices having the finiteness property has an extremal complex polytope norm is false.
Abstract: We disprove a recent conjecture of Guglielmi, Wirth, and Zennaro, stating that any nondefective set of matrices having the finiteness property has an extremal complex polytope norm. We give two counterexamples that show that the conjecture is false even if the set of matrices is supposed to admit the positive orthant as an invariant cone, or even if the set of matrices is assumed to be irreducible.
19 citations
Related Topics (5)
Performance Metrics
| Year | Papers |
|---|---|
| 2013 | 1 |
| 2009 | 3 |
| 2008 | 1 |
| 2007 | 1 |
| 2005 | 2 |
| 2002 | 1 |