Some new bounds on eigenvalues of the Hadamard product and the Fan product of matrices

Let A and B be nonsingular M-matrices. A lower bound on the minimum eigenvalue q(B circle A(-1)) for the Hadamard product of A(-1) and B, and a lower bound on the minimum eigenvalue q(A star B) for the Fan product of A and B are given. In addition, an upper bound on the spectral radius rho(A circle B) of nonnegative matrices A and B is also obtained. These bounds improve several existing results in some cases and the estimating formulas are easier to calculate for they are only depending on the entries of matrices A and B. (C) 2009 Elsevier Inc. All rights reserved.

We are deeply indebted to Cui Guihua for her help in collecting specimens. Thanks are given to I-Shiung Chen and Lee-Shing Fang for providing their paper on S. nantaiensis; and given to Liao Te-yu and Tzeng Chyng-shyan for the information and high quality pictures of S. chapaensis; and also given to Mr. He Yuanhui for translating some Vietnamese literatures into Chinese. The authors are grateful to Yang Jian, Du Lina and Zheng Lanping who helped collecting specimens, taking photos, gathering literatures and improving the manuscript. This work was funded by National Basic Research Program of China (973 Program) (2007CB411600), the National Natural Science Foundation of China (30730017) and China Datang Corporation.