论文标题
等式$ x^\ top ax = b $带有$ b $偏斜的对称：偏斜形式是多少？
The equation $X^\top AX=B$ with $B$ skew-symmetric: How much of a bilinear form is skew-symmetric?
论文作者
论文摘要
给定在$ \ mathbb c^n $上的双线性形式，由矩阵$ a \表示\ mathbb c^{n \ times n} $表示，找到$ \ mathbb c^n $的亚空间的最大维度的问题是，对这个子空间的限制是一个非委托人的限制。最大的可逆偏斜矩阵$ b $，使得方程$ x^\ top ax = b $是一致的（这里$ x^\ top $表示矩阵$ x $的转置）。在本文中，我们通过必要且充分的条件提供一个表征，因为矩阵方程$ x^\ top ax = b $是一致的，当$ b $是一个偏斜的对称矩阵时。对于大多数矩阵$ a \ in \ mathbb c^{n \ times n} $有效。确切地说，条件取决于矩阵$ a $的一致性（CFC）的规范形式，这是三种类型的块的直接总和。该条件对于所有矩阵$ a $都是有效的，除了CFC包含一种类型的块，大小小于$ 3 $的块。但是，我们表明该条件对于所有矩阵$ a $都是必需的。
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 $A$.
