Algebra and Combinatorics

 

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Author(s)
Alexander Kovacec; João Providência; Natalia Bebiano;

Title
On the corners of certain determinantal ranges

Abstract
Let A be a complex n x n matrix and let \SO(n) be the group of real orthogonal of matrices of determinant one. Define \Delta (A)={det(A\comp Q): Q\in \SO(n)}, where \comp denotes the Hadamard product of matrices. For a permutation \sigma on {1,\ldots,n}, define z\sigma=d\sigma(A)=\prodi=1nai\sigma(i). It is shown that if the equation z\sigma=det(A\comp Q) has in \SO(n) only the obvious solutions (Q=(\vei \delta\sigma i, j),\vei=\pm 1 such that \ve1\ldots \ven=\sgn \sigma), then the local shape of \Delta(A) in a vicinity of z\sigma resembles a truncated cone whose opening angle equals z\sigma1\widehat{z\sigma} z\sigma2, where
\sigma1, \sigma2 differ from \sigma by transpositions. This lends further credibility to the well known de Oliveira Marcus Conjecture (OMC) concerning the determinant of the sum of normal n x n matrices. We deduce the mentioned fact from a general result concerning multivariate power series and also use some elementary algebraic topology.

Preprint series
Pré-publicações do Departamento de Matemática da Universidade de Coimbra

Issue
05-11

Year
2005

 
     
 

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