秩剖面矩阵的快速计算与广义Bruhat分解
Fast Computation of the Rank Profile Matrix and the Generalized Bruhat Decomposition
摘要：
行（Resp。列）秩分布描述了矩阵的行（分别为列）梯形。在这里，我们提出了一个新的矩阵不变量，秩配置矩阵，总结所有信息的行和列秩配置文件的所有领导子矩阵。我们表明，这个范式存在，是唯一的任何环，提供的概念，使用的McCoy的秩，在存在零因子。然后，我们探索的条件高斯消除算法计算全部或部分这个不变，通过相应的PLUQ分解。这扩大了计算行或列秩配置文件的已知消除变量集。因此，一个新的克劳特基的情况下的变种显着提高了实际效率，以前已知的实现在有限域。对于秩很小的矩阵，我们还将Storjohann和Yang的技术推广到秩分布矩阵的计算，实现了秩为$r$的$m \times n$矩阵的$(r^ω+mn)^{1+o(1)}$时间复杂度，其中$ω$是矩阵乘法的指数。最后，通过给出Bruhat分解的联系，以及它的几个变型和推广.因此，我们对PLUQ分解的算法改进及其实现直接适用于这些分解。特别是，我们展示了如何PLUQ分解揭示的秩配置矩阵也揭示了一个行和列的输入矩阵或其任何领先的子矩阵的梯队形式，通过一个简单的后处理的行和列排列。
英文摘要：
The row (resp. column) rank profile of a matrix describes the stair-case shape of its row (resp. column) echelon form. We here propose a new matrix invariant, the rank profile matrix, summarizing all information on the row and column rank profiles of all the leading sub-matrices. We show that this normal form exists and is unique over any ring, provided that the notion of McCoy's rank is used, in the presence of zero divisors. We then explore the conditions for a Gaussian elimination algorithm to compute all or part of this invariant, through the corresponding PLUQ decomposition. This enlarges the set of known Elimination variants that compute row or column rank profiles. As a consequence a new Crout base case variant significantly improves the practical efficiency of previously known implementations over a finite field. With matrices of very small rank, we also generalize the techniques of Storjohann and Yang to the computation of the rank profile matrix, achieving an $(r^ω+mn)^{1+o(1)}$ time complexity for an $m \times n$ matrix of rank $r$, where $ω$ is the exponent of matrix multiplication. Finally, by give connections to the Bruhat decomposition, and several of its variants and generalizations. Thus, our algorithmic improvements for the PLUQ factorization, and their implementations, directly apply to these decompositions. In particular, we show how a PLUQ decomposition revealing the rank profile matrix also reveals both a row and a column echelon form of the input matrix or of any of its leading sub-matrices, by a simple post-processing made of row and column permutations.
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