M-矩阵Fan积的特征值下界

Lower Bound on Eigenvalue of the Fan Product of M-matrices

本文利用Brauer卵形定理和Cauchy-Schwitz不等式给出了两个非奇异M-矩阵A和B的Fan积的最小特征值下界的新估计式τ(A★B)≥min i≠j1/2{aiibii+ajjbjj-[(aiibii-ajjbjj)2+4aiibiiajjbjj(ρ2(J(m)A)ρ2(J(m)B))1m]1/2}。此下界估计式比现有几个估计式更为精确。通过数值算例计算得τ(A★B)≥2.783 4,与其他文献中的结果加以比较,表明所得的新估计结果在一定条件下改进了Horn和Johnson给出的经典结果,同时也改进了其他已有的几个结果,比其他结果接近τ(A★B)的真值。

A new lower bound on the minimum eigenvalue for the Fan product of two nonsingular M-matrices A and B is given by using Brauer oval theorem and Cauchy-Schwitz inequality τ（A★B）≥min i≠j 1/2{aiibii＋ajjbjj-[（aiibii-ajjbjj）^2＋4aiibiiajjbjj（ρ^2（J（m）A）ρ^2（J（m）B））1/m]^1/2}.The estimating formulas of the bound is better than several known estimating formulas. By calculating with numerical example, we haver（A★B）≥2. 783 4, compared the new bound with the classical results in the literature, numerical example shows that the new estimating formula improves the result of Horn and Johnson effectively, and also improves the other re- lated results, which approach the real value of τ（A★B） than existing ones in some cases.