We show that the set of $n \times n$ complex symmetric matrix pencils of rank at most $r$ is the union of the closures of $\lfloor r/2\rfloor +1$ sets of matrix pencils with some, explicitly described, complete eigenstructures. As a consequence, these are the generic complete eigenstructures of $n \times n$ complex symmetric matrix pencils of rank at most $r$. We also show that the irreducible components of the set of $n\times n$ symmetric matrix pencils with rank at most $r$, when considered as an algebraic set, are among these closures.

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有界秩的通用对称矩阵铅笔

我们显示，等级最高为$ r $的$ n \ times n $复杂对称矩阵铅笔的集合是$ \ lfloor r / 2 \ rfloor + 1 $矩阵铅笔的闭包与一些明确描述的并集的并集，完整的特征结构。结果，它们是秩最大为$ r $的n次乘以n次对称矩阵铅笔的通用完整特征结构。我们还显示，当被视为代数集时，排名最高为$ r $的$ n \乘以n $对称矩阵铅笔集的不可约成分也属于这些闭包。