非负矩阵最大特征值的上下界估计

Upper and lower bound estimates for the maximum eigenvalue of nonnegative matrices

通过构造两个特殊的矩阵,给出非负矩阵最大特征值的上下界估计式,且这些估计式只和非负矩阵的元素有关,计算方便.同时对非负矩阵最大特征值下界估计式的单调递增性和上界估计式的单调递减性进行了证明.在此基础上,利用单调有界准则证明了上下界估计式极限的存在性.最后将所得结果与经典的Frobenius界值进行比较,数值算例表明估计的有效性和精确性.

By constructing two special matrices,the upper and lower bound estimates of the maximum eigenvalues of the nonnegative matrices are given,and these estimates are only related to the elements of the nonnegative matrices,which are easy to compute.The monotonically increasing property of the lower bound estimates and monotonically decreasing property of the upper bound estimates are also proved.On this basis,the existence of the limits of the upper and lower bound estimates is proved by using the monotone bounded criterion.Finally,the results are compared with the classical Frobenius bounds,and a numerical example is presented to show the validity and accuracy of the estimates.