We show that the set of $m \times m$ complex skew-symmetric matrix polynomials of even grade $d$, i.e., of degree at most $d$, and (normal) rank at most $2r$ is the closure of the single set of matrix polynomials with certain, explicitly described, complete eigenstructure. This complete eigenstructure corresponds to the most generic $m \times m$ complex skew-symmetric matrix polynomials of even grade $d$ and rank at most $2r$. The analogous problem for the case of skew-symmetric matrix polynomials of odd grade is solved in [Linear Algebra Appl., 536:1-18, 2018].

翻译：我们证明，偶次度 $d$（即次数不超过 $d$）且（正规）秩至多为 $2r$ 的 $m \times m$ 复斜对称矩阵多项式集合是某个具有显式描述的完全特征结构的矩阵多项式单集合的闭包。这一完全特征结构对应于偶次度 $d$ 且秩至多 $2r$ 的最通用的 $m \times m$ 复斜对称矩阵多项式。对于奇次度斜对称矩阵多项式的类似问题已在文献 [Linear Algebra Appl., 536:1-18, 2018] 中解决。