On the corners of certain determinantal ranges |
| |
Authors: | Alexander Kovacec Natália Bebiano |
| |
Institution: | a Departamento de Mathemática, Universidade de Coimbra, 3001-454 Coimbra, Portugal b Departamento de Física, Universidade de Coimbra, 3001-454 Coimbra, Portugal |
| |
Abstract: | Let A be a complex n×n matrix and let SO(n) be the group of real orthogonal matrices of determinant one. Define Δ(A)={det(A°Q):Q∈SO(n)}, where ° denotes the Hadamard product of matrices. For a permutation σ on {1,…,n}, define It is shown that if the equation zσ=det(A°Q) has in SO(n) only the obvious solutions (Q=(εiδσi,j),εi=±1 such that ε1…εn=sgnσ), then the local shape of Δ(A) in a vicinity of zσ resembles a truncated cone whose opening angle equals , where σ1, σ2 differ from σ by transpositions. This lends further credibility to the well known de Oliveira Marcus Conjecture (OMC) concerning the determinant of the sum of normal n×n matrices. We deduce the mentioned fact from a general result concerning multivariate power series and also use some elementary algebraic topology. |
| |
Keywords: | 15A15 |
本文献已被 ScienceDirect 等数据库收录! |
|