Corners of certain determinantal ranges
 
 
Description:  Let $A$ be a complex $n\times n$ matrix and let $\SO(n)$ be the group of real orthogonal of matrices of determinannt 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 $z_{\sigma_1} \widehat{z_\sigma} z_{\sigma_2}$, 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 $n\times n$ matrices. We deduce the mentioned fact from a general result concerning multivariate power series and also use some elementary algebraic topology.

Joint paper with
Natália Bebiano$^1$, João de Providência$^2$, Alexander Kova\vcec$^1$,
$^1$Departamento de Mathemática; $^2$Departamento de Física, Univ. Coimbra, 3001-454 Coimbra, Portugal.

bebiano@mat.uc.pt, providencia@teor.fis.uc.pt, kovacec@mat.uc.pt
Area(s):
Date:  2005-06-14
Start Time:   14:30
Speaker:  Alexander Kova\vcec (Universidade de Coimbra)
Place:  Sala 5.5
Research Groups: -Algebra and Combinatorics
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