Extremal case in Marcus-Oliveira conjecture and beyond

Alexander Guterman, Rute Lemos, Graça Soares

Research output: Contribution to journalArticlepeer-review

3 Scopus citations

Abstract

For A, C ∈ Mn the C-determinantal range of A is the following set on the complex plane {up triangle, open}C(A) = {det(A - UCU*): UU* = In}. For normal matrices A and C with eigenvalues α1,..., αn and γ1,..., γn, respectively, Marcus [M. Marcus, Derivations, Plücker relations and the numerical range, Indiana Univ. Math. J. 22 (1973), pp. 1137-1149] and Oliveira [G.N. de Oliveira, Normal matrices (research problem), Linear Multilinear Algebra 12 (1982), pp. 153-154] conjectured that {up triangle, open}C(A) is a subset of the convex hull of the points, σ ∈ Sn, where Sn is the symmetric group of degree n. We investigate the extremal set of matrices for which the equality holds in the Marcus-Oliveira conjecture. We illustrate the use of the obtained results by two different applications. The first one deals with the equality case between the radius of {up triangle, open}C(A) and the radius of the convex hull of the points zσ, σ ∈ Sn. The second one is the characterization of additive Frobenius endomorphisms for the determinantal range or radius on the space Mn and on its real subspace of Hermitian matrices.

Original languageEnglish
Pages (from-to)1206-1222
Number of pages17
JournalLinear and Multilinear Algebra
Volume61
Issue number9
DOIs
StatePublished - 2013
Externally publishedYes

Bibliographical note

Funding Information:
The authors would like to thank the referee for careful reading of the manuscript and the helpful comments. The research contained in this article was started while A. Guterman visited the Department of Mathematics, University of Trás-os-Montes and Alto Douro. He wishes to thank CM-UTAD and the grants RFBR 11-01-00794-a and MD-2502.2012.1 for the financial support. R. Lemos was supported by the Portuguese Foundation for Science and Technology (FCT) through the Center for Research and Development in Mathematics and Applications (CIDMA). G. Soares would like to thank the grant of the Portuguese Government through the FCT under the project PEst-OE/MAT/UI4080/2011.

Keywords

  • Marcus-Oliveira conjecture
  • determinantal range
  • matrix transformations

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