On the estimations of bounds for determinant of Hadamard product of \(H\)-matrices (Q2748454)

Let \(A=\left[ a_{ij}\right] \) and \(B=\left[ b_{ij}\right] \) be \(n\times n\) real matrices, and let \(A\circ B\) denote their Hadamard product. The ``comparison matrix'' \(\mathcal{M}(A)\) of \(A\) is the \(n\times n\) matrix whose \((i,j)\)th entry is \(\left|a_{ii}\right|\) if \(i=j\) and \(-\left|a_{ij}\right|\) if \(i\neq j\). The matrix \(A\) is an nonsingular \(M\)-matrix if it has the form \(sI-C\) where every entry of \(C\) is nonnegative and the spectral radius \(\rho(C)<s\); and \(A\) is an \(H\)-matrix if \(\mathcal{M}(A)\) is a \(M\)-matrix. NEWLINENEWLINENEWLINEThe authors show that if \(A\) and \(B\) are \(H\)-matrices with \(\prod_{i=1}^{n}a_{ii}b_{ii}>0\) then \(\det(A\circ B)\geq \det(\mathcal{M}(A\circ B))>0\). They use this to generalize the inequalities on \(M\)-matrices in the paper of \textit{J. Liu} and \textit{L. Zhu} [SIAM J. Matrix Anal. Appl. 18, No. 2, 305-311 (1997; Zbl 0874.15015)] to \(H\)-matrices.