Norm bounds for summation of two normal matrices

A sharp upper bound is obtained for A+iB , where A and B are n×n Hermitian matrices satisfying a1I A a2I and b1I B b2I. Similarly, an optimal bound is obtained for U+V , where U and V are n×n unitary matrices with any specified spectra; the study leads to some surprising phenomena of discontinuity concerning the spectral variation of unitary matrices. Moreover, it is proven that for two (non-commuting) normal matrices A and B with spectra σ(A) and σ(B), the optimal norm bound for A+B equals Extensionsof the results to infinite dimensional cases are also considered.