|
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):
|
