Abstract view
Nonconvexity of the Generalized Numerical Range Associated with the Principal Character


Published:20001201
Printed: Dec 2000
ChiKwong Li
Alexandru Zaharia
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Abstract
Suppose $m$ and $n$ are integers such that $1 \le m \le n$. For a
subgroup $H$ of the symmetric group $S_m$ of degree $m$, consider
the {\it generalized matrix function} on $m\times m$ matrices $B =
(b_{ij})$ defined by $d^H(B) = \sum_{\sigma \in H} \prod_{j=1}^m
b_{j\sigma(j)}$ and the {\it generalized numerical range} of an
$n\times n$ complex matrix $A$ associated with $d^H$ defined by
$$
\wmp(A) = \{d^H (X^*AX): X \text{ is } n \times m \text{ such that }
X^*X = I_m\}.
$$
It is known that $\wmp(A)$ is convex if $m = 1$ or if $m = n = 2$.
We show that there exist normal matrices $A$ for which $\wmp(A)$ is
not convex if $3 \le m \le n$. Moreover, for $m = 2 < n$, we prove
that a normal matrix $A $ with eigenvalues lying on a straight line
has convex $\wmp(A)$ if and only if $\nu A$ is Hermitian for some
nonzero $\nu \in \IC$. These results extend those of Hu, Hurley
and Tam, who studied the special case when $2 \le m \le 3 \le n$
and $H = S_m$.