The problem of extracting a well conditioned submatrix from any rectangular matrix (with normalized columns) has been studied for some time in functional and harmonic analysis. In the seminal work of Bourgain and Tzafriri and many subsequent improvements, methods using random column selection were considered. Constructive approaches have been proposed lately, mainly sparked by the work of Batson, Spielman and Srivastava. The column selection problem we consider in this paper is concerned with extracting a well conditioned submatrix, and more precisely, a matrix with all its singular values being contained in the interval $[1-\epsilon ,1+\epsilon ]$. Such results are known to have far reaching connections with many fields in mathematics and engineering.

Our main contribution is a new deterministic method that achieves the same order $R$ for the number of selected columns as in Bourgain and Tzafriri’s original Theorem, up to a $log\left(R\right)$ multiplicative factor. Our analysis is elementary and shows how a simple eigenvalue perturbation argument can lead to an intuitive and very short proof. We also obtain individual lower and upper bounds for each singular value of the extracted matrix.

在功能和谐波分析中，已经研究了从任何矩形矩阵（具有标准化列）中提取条件良好的子矩阵的问题。在Bourgain和Tzafriri的开创性工作以及许多后续改进中，考虑了使用随机列选择的方法。最近提出了建设性的方法，主要是由Batson，Spielman和Srivastava的工作激发的。我们在本文中考虑的列选择问题与提取条件良好的子矩阵有关，更确切地说，是一个矩阵，其中所有奇异值都包含在区间中$[\mathrm{1个}-\epsilon ，\mathrm{1个}+\epsilon ]$。众所周知，这样的结果与数学和工程学的许多领域都有深远的联系。

我们的主要贡献是实现相同顺序的新确定性方法 $\mathrm{[R}$ 根据布尔加因（Bourgain）和扎夫里里（Tzafriri）的原始定理，选择的列数最多 $日志（\mathrm{[R}）$乘法因子。我们的分析是基本的，它显示了一个简单的特征值摄动参数如何导致直观且简短的证明。我们还为提取的矩阵的每个奇异值获取了单独的上下限。