方程$X^\top AX=B$与$B$反对称：一个双线性型有多少是反对称的？
The equation $X^\top AX=B$ with $B$ skew-symmetric: How much of a bilinear form is skew-symmetric?
摘要：
给定$\mathbb C^n$上的双线性型，由矩阵$A\in\mathbb C^{n\times n}$表示，找到$\mathbb C^n$的子空间的最大维数使得$A$对该子空间的限制是非退化反对称双线性型的问题等价于找到最大可逆反对称矩阵$B$的大小使得方程$X^\top AX=B$是相容的（这里$X^\top$表示矩阵$X$的转置）。本文给出了矩阵方程$X^\top AX=B$在$B$是反对称矩阵时相容的一个充分必要条件.此条件对大多数矩阵$A\in\mathbb C^{n\times n}$都有效。确切地说，该条件取决于矩阵$A$的同余标准形（CFC），其是三种类型的块的直和。该条件对所有矩阵$A$都有效，但CFC包含块的矩阵除外，块的类型之一，大小小于$3$。然而，我们表明，该条件是必要的所有矩阵$A$。
英文摘要：
Given a bilinear form on $\mathbb C^n$, represented by a matrix $A\in\mathbb C^{n\times n}$, the problem of finding the largest dimension of a subspace of $\mathbb C^n$ such that the restriction of $A$ to this subspace is a non-degenerate skew-symmetric bilinear form is equivalent to finding the size of the largest invertible skew-symmetric matrix $B$ such that the equation $X^\top AX=B$ is consistent (here $X^\top$ denotes the transpose of the matrix $X$). In this paper, we provide a characterization, by means of a necessary and sufficient condition, for the matrix equation $X^\top AX=B$ to be consistent when $B$ is a skew-symmetric matrix. This condition is valid for most matrices $A\in\mathbb C^{n\times n}$. To be precise, the condition depends on the canonical form for congruence (CFC) of the matrix $A$, which is a direct sum of blocks of three types. The condition is valid for all matrices $A$ except those whose CFC contains blocks, of one of the types, with size smaller than $3$. However, we show that the condition is necessary for all matrices $A$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
