Nonconvexity of the Generalized Numerical Range Associated with the Principal Character

Canadian mathematical bulletin

关键词: convexity;    generalized numerical range;    matrices;   

DOI  :  10.4153/CMB-2000-053-3

学科分类：数学（综合）

来源: University of Toronto Press * Journals Division

【 摘 要 】

Suppose $m$ and $n$ are integers such that $1 le m le n$. For asubgroup $H$ of the symmetric group $S_m$ of degree $m$, considerthe {it generalized matrix function} on $mimes m$ matrices $B =(b_{ij})$ defined by $d^H(B) = sum_{sigma in H} prod_{j=1}^mb_{jsigma(j)}$ and the {it generalized numerical range} of an$nimes n$ complex matrix $A$ associated with $d^H$ defined by $$wmp(A) = {d^H (X^*AX): X ext{ is } n imes m ext{ such that } X^*X = I_m}.$$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 $wmp(A)$ isnot convex if $3 le m le n$. Moreover, for $m = 2 < n$, we provethat a normal matrix $A $ with eigenvalues lying on a straight linehas convex $wmp(A)$ if and only if $u A$ is Hermitian for somenonzero $u in IC$. These results extend those of Hu, Hurleyand Tam, who studied the special case when $2 le m le 3 le n$and $H = S_m$.

【 授权许可】

Unknown