关于除环上分块矩阵的Marsaglia-Styan秩公式的注记

Notes on the Marsaglia-Styan Rank Formula of Partitioning Matrix over a Division Ring

指出已有文献中的除环上2×2分块矩阵ABC:D的Marsaglia-Styan秩公式和数域上的表达形式是相同的,即其表达式中三处出现的A的:1:-广义逆都是相同的。应用除环上的初等变换的方法,证明了分块矩阵的Marsaglia-Styan秩公式的表达式中的三处也可以选择不同的A的:1:-广义逆。

In all the available articles, the expression form of Marsaglia-Styan rank formula of 2×2 partitioning matrix [ABCD]over division rings is the same as it over number fields, that is, the three [1]-generalized inverses of A appearing in the formula are the same. In this paper, by using the method of elementary transformation of division rings, we prove that the three [1]-generalized inverses of A appearing in the formula of Marsaglia-Styan rank formula on partitioning matrix can be different.