In this paper, we are interested in the eigenproblem on the large and low-rank matrix $S=A{B}^H$, where $A,B\in {ℂ}^{n\times r}$ are of full column rank and $r\ll n$. To the best of our knowledge, there are no results on the relations between the Jordan decomposition and the Schur decomposition of $B^{H}A$ and those of $A{B}^H$. Some known results are only on characteristic polynomials, elementary divisors, and Jordan blocks of $A{B}^H$, and are purely theoretical and are not easy to use for computational purposes. Based on the Jordan decomposition and the Schur decomposition of the small matrix $B^{H}A\in {ℂ}^{r\times r}$, we consider how to derive those of the large matrix $A^{H}B\in {ℂ}^{n\times n}$ in this work. The construction methods proposed are not only theoretical but also practical. Numerical experiments show the effectiveness of our theoretical results.

在本文中，我们对大和低秩矩阵的本征问题感兴趣 $\mathrm{小号}=\mathrm{一种}{乙}^H$，在哪里 $\mathrm{一种}，乙\in {ℂ}^{ñ\times \mathrm{[R}}$ 具有完整的列级并且 $\mathrm{[R}\ll ñ$。据我们所知，尚无关于约旦分解和舒尔分解的关系的结果。${乙}^H\mathrm{一种}$ 和那些 $\mathrm{一种}{乙}^H$。一些已知的结果仅针对特征多项式，基本除数和$\mathrm{一种}{乙}^H$，而且纯粹是理论性的，不易用于计算目的。基于小矩阵的约旦分解和舒尔分解${乙}^H\mathrm{一种}\in {ℂ}^{\mathrm{[R}\times \mathrm{[R}}$，我们考虑如何导出大矩阵的那些${\mathrm{一种}}^H乙\in {ℂ}^{ñ\times ñ}$在这项工作中。提出的施工方法不仅是理论上的，而且是实际的。数值实验表明了我们理论结果的有效性。