Some new bounds on eigenvalues of the Hadamard product and the Fan product of matrices
Let \(A\) and \(B\) be nonsingular \(M\)-matrices. The authors obtain lower bounds on the minimum eigenvalues of the Hadamard product \(B\circ A^{-1}\), and of the Fan product \(A\star B\), where the \(i,j\) entry of the former is \(a_{i,j}b_{i,j}\), and of the latter is \(a_{i,j}b_{i,j}\) if \(i=j\) and \(-a_{i,j}b_{i,j}\) if \(i\neq j\). These bounds improve the recent estimates by \textit{M. Fang} [Linear Algebra Appl. 425, No.~1, 7--15 (2007; Zbl 1128.15011)] and \textit{R. Huang} [Linear Algebra Appl. 428, No.~7, 1551--1559 (2008; Zbl 1163.15017)]. Unfortunately, the authors ignore some estimations presented recently in the area due to \textit{D. Kun} [J. East China Norm. Univ., Nat. Sci. Ed. 2008, No.~5, 45--50 (2008; Zbl 1199.15069)] and to \textit{Q. Liu} and \textit{G. Chen} [Linear Algebra Appl. 431, No.~5--7, 974--984 (2009; Zbl 1183.15017)], for example.
scientific article; zbMATH DE number 5649993
scientific article