Lower bounds for the smallest singular value via permutation matrices (Q6570476)

The authors start by recalling some known lower bounds for the smallest singular value \(\sigma_n(A)\) of a matrix \(A\). Then they explain the application of permutation matrices in order to improve the following bound given in [\textit{C. R. Johnson}, Linear Algebra Appl. 112, 1--7 (1989; Zbl 0723.15013)].\NFor an \(n\times m\) matrix \(A=(a_{i,j})\) with \(n\le m\), assuming the singular values are arranged in decreasing order, the singular value \(\sigma_n(A)\) satisfies\N\[\N\sigma_n(A)\ge \underset{1\le k\le n}{\min}\left\{|a_{k,k}|-\frac{1}{2}\left[\sum_{i=1, i\neq k}^n|a_{k,i}|+\sum_{i=1,i\neq k}^n|a_{i,k}|\right]\right\}.\N\]