On a bound for the Hadamard product of an M-matrix and its inverse (Q802699)

For an \(n\times n\) nonsingular \(M-matrix\) A, let \(q(A\circ A^{-1})\) denote the smallest real eigenvalue of the Hadamard product of A with its inverse \(A^{-1}\). For \(n>2\), it is shown that \(q(A\circ A^{-1})>2/n\) and in fact the inequality is sharp. If \(n=2\), then \(q(A\circ A^{- 1})=1\). This proves a conjecture of \textit{M. Fiedler} and \textit{T. L. Markham} [ibid. 101, 1-8 (1988; Zbl 0648.15009)].