Maximizing the numerical radii of matrices by permuting their entries

† Dedicated to Professor Pei Yuan Wu.

Abstract

Let be an complex matrix such that every row and every column has at most one non-zero entry. We determine permutations of the non-zero entries of so that the resulting matrix has maximum numerical radius. Extension of the results to operators acting on separable Hilbert spaces are also obtained. Related results and additional problems are also mentioned.

Keywords:

• numerical range and radius
• adjacency matrices
• paths
• trees

AMS Subject Classification:

• 15A60

Acknowledgments

The authors would like to thank the referee for many helpful suggestions and careful reading of the paper, and thank Professor Hwa-Long Gau for some valuable comments.

This research was done while Li was visiting the University of Hong Kong in the Spring of 2012. His research was supported by a USA NSF grant, a HK RCG grant, and the Shanxi Hundred Talent Scholar program. He is an honorary professor of the Shanghai University.

Cheung is a research associate of the University of Hong Kong.

Notes

† Dedicated to Professor Pei Yuan Wu.