Improved Error Bounds for the Eigenvalues of Certain Normal Operators
James Alan Cochran, Erold W. Hinds · SIAM Journal on Numerical Analysis · 1972
In 1953 Hoffman and Wielandt proved that if A and B are normal matrices of finite order n with eigenvalues $\lambda _ u ,\mu _ u ,1 \leqq u \leqq n$, respectively, then there exists a suitable ordering of the eigenvalues such that $\Sigma _{ u = 1}^n |\lambda _ u - \mu _ u |^2 \leqq \|A - B\|_e^2 $, where $\| \cdot \|_e $ is the usual Euclidean matrix norm. In this paper, the Hoffman–Wielandt result is extended to the case of completely continuous normal operators of Hilbert–Schmidt type. Other generalized inequalities which are useful in the numerical determination of eigenvalue approximations associated with such operators are also considered herein.