TITLE: Changing the spectrum of a symmetric matrix by diagonal perturbations
QUESTION [9 upvotes]: Given a fixed symmetric matrix $S$, can one change the spectrum of $S$ to any desired set of eigenvalues $\{\lambda_1,\dots,\lambda_n\}$ by adding a diagonal matrix $D$ to $S$?

REPLY [2 votes]: Schmuel Friedland proves in "Matrices with prescribed off-diagonal elements" 
(link)
that, given M any n-by-n matrix over $\mathbb{C}$, there exists a diagonal matrix (also taken over $\mathbb{C}$ ) such that $M+D$ has exactly the eigenvalues you want.
I don't know of any related results specific to M an n-by-n real symmetric matrix.