Given a square pencil $A+ \lambda B$, where $A$ and $B$ are complex matrices, we consider the problem of finding the singular pencil nearest to it in the Frobenius distance. This problem is known to be very difficult, and the few algorithms available in the literature can only deal efficiently with pencils of very small size. We show that the problem is equivalent to minimizing a certain objective function over the Riemannian manifold $SU(n) \times SU(n)$, where $SU(n)$ denotes the special unitary group. With minor modifications, the same approach extends to the case of finding a nearest singular pencil with a specified minimal index. This novel perspective is based on the generalized Schur form of pencils, and yields a competitive numerical method, by pairing it with an algorithm capable of doing optimization on a Riemannian manifold. We provide numerical experiments that show that the resulting method allows us to deal with pencils of much larger size than alternative techniques, yielding candidate minimizers of comparable or better quality. In the course of our analysis, we also obtain a number of new theoretical results related to the generalized Schur form of a (regular or singular) square pencil and to the minimal index of a singular square pencil whose nullity is $1$.

翻译：考虑方束 $A+ \lambda B$，其中 $A$ 和 $B$ 为复矩阵，问题旨在寻找其在Frobenius距离下的最近奇异束。该问题已知非常困难，现有文献中的少数算法仅能有效处理极小尺寸的束。我们证明该问题等价于在黎曼流形 $SU(n) \times SU(n)$（$SU(n)$ 表示特殊酉群）上最小化某一目标函数。通过微小修改，同一方法可推广至寻找具有指定最小指标的最近奇异束。这一新颖视角基于束的广义Schur型，结合能在黎曼流形上进行优化的算法，形成了一种有竞争力的数值方法。数值实验表明，该方法能处理远大于其他技术的束尺寸，且得到的候选极小化子质量相当或更优。在分析过程中，我们还获得了若干与（正则或奇异）方束的广义Schur型及零度为1的奇异方束最小指标相关的新理论结果。