TITLE: Eigenvalues of Symmetric Matrix Plus Diagonal Matrix
QUESTION [7 upvotes]: Let $M$ be a real symmetric matrix and let $D$ be a real diagonal matrix. 

Can we say something about the eigenvalues of $M+D$ in terms of the
  eigenvalues of $M$?

For example, if $D$ is a constant multiple of $I$ ($D = cI$), then the eigenvalues of $M+D$ is just $c$ plus the eigenvalues of $M$. Can we get bounds on the locations of the eigenvalues of $M+D$ ?

REPLY [6 votes]: Your question is equivalent to this one 
let $A,B$ be symmetric real matrices, the spectra of which, are known. What can be said about $spectrum(A+B)$ ?
This question was the subject of Horn's conjecture that lists all the possible values of $spectrum(A+B)$.
This very difficult conjecture was proved in 1999 by Klyachko, Knutson and the inevitable Tao.
cf. this paper by Bhatia and the references inside.
http://math.univ-lyon1.fr/~ressayre/PDFs/bhatia.pdf