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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)=\prod_{i=1}^{n}a_{i\sigma(i)}. It is shown that if the equation z_{\sigma}=det(A\comp Q) has in \SO(n) only the obvious solutions (Q=(\ve_{i} \delta_{\sigma i, j}),\ve_{i}=\pm 1 such that \ve_{1}\ldots \ve_{n}=\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
\sigma_{1}, \sigma_{2} 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 Prépublicações do Departamento de Matemática da Universidade de Coimbra
Issue 0511
Year 2005
