We investigate the structure of the multiplicative semigroup generated by the set of matrices that are unitarily equivalent to a given invertible matrix A. In particular, we give necessary and sufficient conditions for such a semigroup to be the special linear group or the general linear group or other classical matrix groups. Furthermore, we use the above to determine the subgroups of the general linear group that are normalized by the unitary group.

中文翻译：

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单一酉轨道产生的群和半群

我们研究由与给定的可逆矩阵 A 酉等价的矩阵集合产生的乘法半群的结构。特别地，我们给出了这样一个半群是特殊线性群或一般线性群的充要条件或其他经典矩阵群。此外，我们使用上面的来确定一般线性群的子群，这些子群被酉群归一化。