Canadian Journal of Mathematics-Journal Canadien DE Mathematiques
Let A and B be n x n complex Hermitian (or real symmetric) matrices with eigenvalues a(1) >= ... >= a(n) and b(1) >= ... >= b(n). All possible inertia values, ranks, and multiple eigenvalues of A + B are determined. Extension of the results to the sum of k matrices with k > 2 and connections of the results to other subjects such as algebraic combinatorics are also discussed.
Li, C. K., & Poon, Y. T. (2010). Sum of Hermitian matrices with given eigenvalues: inertia, rank, and multiple eigenvalues. Canadian Journal of Mathematics, 62(1), 109-132.