Eigenvalues of sum of a Hermitian matrix and a diagonal matrix

[peterlam]

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!

 

[Some Pig]

c₁+c₂+...+c_n=a₁+a₂+...+a_n+b₁+b₂+...+b_n
min{c₁,c₂,...,c_n} ≤ (a₁+a₂+...+a_n+b₁+b₂+...+b_n)/n