We improve the current best running time value to invert sparse matrices over finite fields, lowering it to an expected $O\big(n^{2.2131}\big)$ time for the current values of fast rectangular matrix multiplication. We achieve the same running time for the computation of the rank and nullspace of a sparse matrix over a finite field. This improvement relies on two key techniques. First, we adopt the decomposition of an arbitrary matrix into block Krylov and Hankel matrices from Eberly et al. (ISSAC 2007). Second, we show how to recover the explicit inverse of a block Hankel matrix using low displacement rank techniques for structured matrices and fast rectangular matrix multiplication algorithms. We generalize our inversion method to block structured matrices with other displacement operators and strengthen the best known upper bounds for explicit inversion of block Toeplitz-like and block Hankel-like matrices, as well as for explicit inversion of block Vandermonde-like matrices with structured blocks. As a further application, we improve the complexity of several algorithms in topological data analysis and in finite group theory.

中文翻译：

---

有限域中更快的稀疏矩阵求逆和秩计算

我们改进了当前最佳运行时间值以在有限域上反转稀疏矩阵，将其降低到快速矩形矩阵乘法的当前值的预期 $O\big(n^{2.2131}\big)$ 时间。对于有限域上稀疏矩阵的秩和零空间的计算，我们实现了相同的运行时间。这种改进依赖于两个关键技术。首先，我们采用将任意矩阵分解为 Eberly 等人的块 Krylov 和 Hankel 矩阵。(ISSAC 2007)。其次，我们展示了如何使用结构化矩阵的低位移秩技术和快速矩形矩阵乘法算法来恢复块 Hankel 矩阵的显式逆。我们将我们的反演方法推广到具有其他位移算子的块结构矩阵，并加强了块 Toeplitz-like 和块 Hankel-like 矩阵的显式反演以及具有结构块的块 Vandermonde-like 矩阵的显式反演的最著名的上限. 作为进一步的应用，我们提高了拓扑数据分析和有限群论中几种算法的复杂性。