关于Wielandt-Hoffman定理

关于正规矩阵的任意扰动,有下述定理成立. 定理1.设A为n阶正规矩阵,C为n阶任一矩阵.A的特征值为λ_1,…,λ_n,C的特征值为μ_1…,μ_n.C~H表示C的转置共轭,||·||_2与||·||_F分别表示矩阵的谱范数与Frobenius范数.记

ON THE WIELANDT-HOFFMAN THEOREM

In this note, using the departure from normality of matrix we obtain a upper boundof "distance" between respective eigenvalues of a normal matrix and an arbitrary ma-trix. Moreover we obtain a easily computed upper bound of this "distance" in termsof the upper bound of the departure from normality which has been given by Henriciand the lower bound of this departure which is given in this note. We also give aelementary and simple prove of the Wielandt-Hoffman theorem for any Hermitianmatrices.