# Eigenvalues of sum of a Hermitian matrix and a diagonal matrix

Consider two matrices:
1) A is a n-by-n Hermitian matrix with real eigenvalues a_1, a_2, ..., a_n;
2) B is a n-by-n diagonal matrix with real eigenvalues b_1, b_2, ..., b_n.

If we form a new matrix C = A + B, can we say anything about the eigenvalues of C (c_1, ..., c_n) from the eigenvalues of A and B? Can we determine c_1, ..., c_n from a_1, ..., a_n, b_1, ..., b_n? If not, can we just determine the smallest eigenvalue of C from A and B?

Thank you!