Comparison of infinity norms of matrix inverses (Q1109851)

The author studies matrices for which addition of positive numbers to the diagonal cause nonincreasing changes in the \(\ell_{\infty}\) norm of rows of its inverse. He proves that if T is a matrix having an inverse with positive diagonal and no zero off diagonal entries then \(T=D^{- 1}SD\), where D is a diagonal matrix with the diagonal entries \(+1\) or -1 and S has a positive inverse. Another result concerns matrices with a nonnegative diagonal of the inverse.