An upper bound for \(\| A^{-1}\|_{\infty}\) of strictly diagonally dominant \(M\)-matrices (Q996319)

A square \(n\times n\) matrix \(A\) is called a nonsingular \(M\)-matrix if there exists an \(n\times n\) nonnegative matrix \(P\) such that \(A=sI-P,\) where \(I\) is the identity matrix and \(s>\rho(P)\), \(\rho(P)\) is the spectral radius of the matrix \(P\). It is clear that \(A^{-1}\) is a nonnegative matrix. In this paper, the upper bound of \(\| A^{-1}\|_\infty\) is improved and a new lower bound of \(\rho(A^{-1})^{-1}\) is obtained where the Perron eigenvalue of \(A^{-1}\) is denoted by \(\rho(A^{-1})\). Here it is necessary to point out that the authors in the abstract of this paper say that they give a sharp upper bound for \(\| A^{-1}\|_\infty\), this is not correct (see Example 3.1 of the paper).