矩阵Hadamard积最小特征值的新界值估计

New Bound on the Minimum Eigenvalue of the Hadamard Product of Matrices

设A和B是非奇异M矩阵,给出B和A-1的Hadamard积的最小特征值的新界值估计,设矩阵A=(aij)和B=(bij)都为非奇异M矩阵,A-1=(βij),则有τ(BA-1)≥min i≠j12{βiibii+βjjbjj-[(βiibii-βjjbjj)2+4sisjβiiβjj(bii-τ(B))(bjj-τ(B))]12}。估计式仅依赖矩阵的元素,易于计算。数值例子表明所得新估计式改进了现有的一些结果。

If A and B are nonsingular M matrices. A new bound on the minimum eigenvalue of the Hadamard product for B and A-1 is given. LetA=（ailj）and B= （bij）are nonsingular M matrices,A 1 = （βij）. We have r（B°A^-1）≥ min i≠j1/2{βiibii＋βjjbjj-[（βiibii-βjjbjj）^2＋4sisjβiiβjj（bii-τ（B））（bjj-τ（B））]^1/2}. The bound is easier to calculate since they only depend on the entries of matrices. Finally, the numerical example shows that the bound improves some estimating results.