If two Hermitian matrices commute, then the eigenvalues of their sum are just the sums of the eigenvalues of the two matrices in a suitable order. Examples show that the converse is not true in general. In this paper, partial converses are obtained. The technique involves a characterization of the equality cases for Weyl's inequalities. Moreover, a new proof on the commutativity of two Hermitian matrices with property L and analogous results for the product of two positive definite Hermitian matrices are included. © 1994.
So, W. (1994). Commutativity and spectra of Hermitian matrices. Linear Algebra and Its Applications, 212–213(C), 121–129. https://doi.org/10.1016/0024-3795(94)90399-9