On invertibility and positive invertibility of matrices (Q5935364)

Let \(A=[a_{ij}]\) be an \(n\times n\) complex matrix whose diagonal entries are all nonzero. For each \(k\) define NEWLINE\[NEWLINE \tilde{v}_{k}:=\max\left\{\left|a_{jk}\right|:j=1,\dots ,k-1\right\} \text{ and }\tilde{w}_{k}:=\max\left\{ \left|a_{jk}\right|:j=k+1,\dots ,n-1\right\} NEWLINE\]NEWLINE and put NEWLINE\[NEWLINE m:=\prod_{k=2}^{n}\left( 1+\frac{\tilde{v}_{k}}{\left|a_{kk}\right|}\right) \text{ and }m':=\prod_{k=1}^{n-1}\left( 1+\frac{\tilde {w}_{k}}{\left|a_{kk}\right|}\right) NEWLINE\]NEWLINE Suppose that \(mm'<m+m'\). Then the author proves that \(A\) is invertible and NEWLINE\[NEWLINE \left\|A^{-1}\right\|_{\infty}\leq\frac{mm'}{(m+m-mm')l} NEWLINE\]NEWLINE where \(l:=\min\{\left|a_{kk}\right|:k=1,\dots ,n\}.\) This result strengthens similar criteria due to Levy-Desplanques and \textit{L. Collatz} [Functional analysis and numerical mathematics (1966; Zbl 0148.39002), p. 377].