论文标题
在矩阵方程的一致性上
On the consistency of the matrix equation $X^\top A X=B$ when $B$ is symmetric: the case where CFC($A$) includes skew-symmetric blocks
论文作者
论文摘要
在本文中，这是[A. Borobia，R。Canogar，F。DeTerán，Mediterr。 J. Math。 18，40（2021）]，我们为矩阵方程提供了必要且充分的条件，$ x^\ top ax = b $是在对称的$ b $时保持一致的。条件取决于矩阵$ a $的一致性的规范形式，因此被证明是所有矩阵$ a $所必需的，并且对于大多数矩阵来说就足够了。该结果在上一篇论文中改善了主要论文，因为该条件比该参考文献中的条件更强，并且可以保证一组较大的矩阵（即，那些适合一致性的规范形式的矩阵，CFC（$ a $）包括偏斜的合成块）。
In this paper, which is a follow-up to [A. Borobia, R. Canogar, F. De Terán, Mediterr. J. Math. 18, 40 (2021)], we provide a necessary and sufficient condition for the matrix equation $X^\top AX=B$ to be consistent when $B$ is symmetric. The condition depends on the canonical form for congruence of the matrix $A$, and is proved to be necessary for all matrices $A$, and sufficient for most of them. This result improves the main one in the previous paper, since the condition is stronger than the one in that reference, and the sufficiency is guaranteed for a larger set of matrices (namely, those whose canonical form for congruence, CFC($A$), includes skew-symmetric blocks).
