完全非负矩阵上的Oppenheim不等式的推广

The Generalization of Oppenheim's Inequalities for Totally Nonnegative Matrices

完全非负矩阵在Hadamard乘积意义下是不封闭的。对于两个三对角完全非负矩阵A=(a_(ij)),B=(b_(ij)),Markham证明了它们的Hadamard乘积的行列式满足Oppenheim不等式。我们应用完全非负矩阵的Hadamard中心的性质,改进了Markham的相应结果,给出了新的下界(A_1为删去第一行的A的主子矩阵):det(AB)≥(multiply from i=1 to n b_(ii))detA+(multiply from i=1 to n a_(ii))detB-detAdetB+(detA)((multiply from i=2 to n a_(ii)/detA_1)-1)(b_(11)detB_1-detB)+(detB)((multiply from i=2 to n b_(ii)/detB_1)-1)(a_(11)detA_1-detA)。

Totally nonnegative matrices are not closed in the sense of Hadamard product. For two tridiag-onal totally nonnegative matrices A = (aij ) ,B =(bij), Markham proved that the determinant of their Hadamard product satisfies Oppenheim's inequality. Applying the properties of Hadamard core for totally non-negative matrices, we obtain the new lower bounds of the determinant about Hadamard product for two tridiagonal totally nonnegative matricies,and improve the corresponding results derived by Markham. Our conclusion is as follows: where A1 is principal submatrix of a obtained by deleting its first row and column.