论文标题
基质空间，基质组和多线性图的一致性
Congruence of matrix spaces, matrix tuples, and multilinear maps
论文作者
论文摘要
两个矩阵向量空间$ v，w \ subset \ mathbb c^{n \ times n} $，如果某些非单词$ s $和$ r $的$ svr = w $，则是等效的。如果$ r = s^t $，这些空间是一致的。我们证明，如果$ v $和$ w $中的所有矩阵都是对称的，或者$ v $中的所有矩阵和$ w $都是偏斜的对称，那么$ v $和$ w $是一致的，并且仅当它们等效时。令$ f：u \ times \ dots \ times u \ to v $和$ g：u'\ times \ dots \ times \ times u'\ to v'$是对称的或偏斜的$ k $ - $ k $ - linear maps abo $ \ mathbb c $。如果存在一组线性将$φ_1，\ dots，φ_k：u \ to u'$和$ψ：v \ to v'$，将$ f $转换为$ g $，则存在$φ_1= \ dots =φ_k$。
Two matrix vector spaces $V,W\subset \mathbb C^{n\times n}$ are said to be equivalent if $SVR=W$ for some nonsingular $S$ and $R$. These spaces are congruent if $R=S^T$. We prove that if all matrices in $V$ and $W$ are symmetric, or all matrices in $V$ and $W$ are skew-symmetric, then $V$ and $W$ are congruent if and only if they are equivalent. Let $F: U\times\dots\times U\to V$ and $G: U'\times\dots\times U'\to V'$ be symmetric or skew-symmetric $k$-linear maps over $\mathbb C$. If there exists a set of linear bijections $φ_1,\dots,φ_k:U\to U'$ and $ψ:V\to V'$ that transforms $F$ to $G$, then there exists such a set with $φ_1=\dots=φ_k$.
