The concept of matrix rigidity was introduced by Valiant(independently by Grigoriev) in the context of computing linear transformations. A matrix is rigid if it is far(in terms of Hamming distance) from any matrix of low rank. Although we know rigid matrices exist, obtaining explicit constructions of rigid matrices have remained a long-standing open question. This decade has seen tremendous progress towards understanding matrix rigidity. In the past, several matrices such as Hadamard matrices and Fourier matrices were conjectured to be rigid. Very recently, many of these matrices were shown to have low rigidity. Further, several explicit constructions of rigid matrices in classes such as $E$ and $P^{NP}$ were obtained recently. Among other things, matrix rigidity has found striking connections to areas as disparate as communication complexity, data structure lower bounds and error-correcting codes. In this survey, we present a selected set of results that highlight recent progress on matrix rigidity and its remarkable connections to other areas in theoretical computer science.

中文翻译：

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矩阵刚性的最新进展——调查

矩阵刚性的概念由 Valiant（由 Grigoriev 独立）在计算线性变换的上下文中引入。如果矩阵远离任何低秩矩阵（就汉明距离而言），则该矩阵是刚性矩阵。尽管我们知道刚性矩阵存在，但获得刚性矩阵的显式构造仍然是一个长期存在的悬而未决的问题。这十年在理解矩阵刚性方面取得了巨大进步。过去，Hadamard 矩阵和 Fourier 矩阵等几种矩阵都被推测为刚性矩阵。最近，许多这些矩阵被证明具有低刚性。此外，最近还获得了几个类中刚性矩阵的显式构造，例如 $E$ 和 $P^{NP}$。除其他外，矩阵刚性已经发现与沟通复杂性等不同领域的惊人联系，数据结构下界和纠错码。在本次调查中，我们展示了一组选定的结果，重点介绍了矩阵刚性的最新进展及其与理论计算机科学其他领域的显着联系。