Partial Superdiagonal Elements and Singular Values of a Complex Skew-Symmetric Matrix
1998; Society for Industrial and Applied Mathematics; Volume: 19; Issue: 3 Linguagem: Inglês
10.1137/s0895479896312559
ISSN1095-7162
Autores Tópico(s)Advanced Combinatorial Mathematics
ResumoLet A be an m x m complex skew-symmetric matrix with singular values $s_1\ge s_1 \ge s_2 \ge s_2\ge \cdots \ge s_n\ge s_n \ge 0$, where n = [m/2]$. We consider the sets ${\D}_p(A) = {diag (UTAU[1,..., p|n+1, ..., n+p]): U\in U(m)\}$, $p = 1, \dots , n$, where $U(m)$ denotes the unitary group. We prove that when m=2n and p=n, d = (d1 ,..., dn) \in {\D}_n(A)$ if and only if \begin{eqnarray*} \sum_{i=1}^k |d_i|&\le &\sum_{i=1}^k s_i, \qquad k = 1, \dots , n,\\ \sum_{i=1}^{n-1} |d_i| - |d_n| &\le &\sum_{i=1}^{n-1} s_i - s_n, \end{eqnarray*} after rearranging the entries of d in descending order with respect to absolute value. The set is not convex in general. The inequalities are identical to those of Thompson--Sing's theorem on the diagonal elements and the singular values of an n x n complex matrix. All other cases, i.e., (1) $m=2n+1$ and $1\le p\le n$ and (2) $m=2n$ and $1\le p < n$, are completely described by the inequailities $\sum_{i=1}^k |d_i|\le \sum_{i=1}^k s_i$, $k = 1, \dots , p$. The sets are all convex. Various applications and related results are obtained.
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