Changing the spectrum of a symmetric matrix by diagonal perturbations 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 $S$?
 A: Let $A$ be a given symmetric matrix over the reals and let $B$ be
a symmetric matrix over the reals as well that we add to $A$ to obtain
$\tilde{A}=A+B$. Let $B=S \Lambda_B S^{-1}$ be the eigendecomposition of
$B$. Then $\tilde{A} = S\left(S^{-1}AS + \Lambda_B\right)S^{-1}$. From that
we see that the eigenvalues of $\tilde{A}$ are precisely the eigenvalues of
$S^{-1}AS + \Lambda_B$. Since the eigenvalues of $S^{-1}AS$ are identical to
those of $A$, we see that, for the purpose of spectrum modification, we lose no
generality if we perturb $A$ with a diagonal matrix initially. 
So let's consider $\tilde{A}=A+B$, where $B$ is diagonal.
Assuming no specific structure on $A$, the answer to the question "what is 
the spectrum of $\tilde{A}$" is not known, to the best of my knowledge. Instead,
there exist some interesting theorems that give information about the distribution of the spectrum of $\tilde{A}$ with respect to that of $A$ and $B$.
The ones more interesting to me, are two theorems by the great Hermann Weyl,
known as Weyl 1 and Weyl 2. Let the eigenvalues of $A$ be ordered as 
$\lambda_1(A) \le \lambda_2(A) \cdots \le \lambda_n(A)$ and similarly for the other matrices. Then 
(Weyl 1)
$\lambda_k(A)+\lambda_1(B) \le \lambda_k(A+B) \le \lambda_k(A) + \lambda_n(B)$
and 
(Weyl 2)
$\lambda_{j+k-n}(A+B) \le \lambda_j(A) + \lambda_k(B) \le \lambda_{j+k-1}(A+B)$
where we interpret an eigenvalue corresponding to an index greater than n as plus infinity and less than 1 as minus infinity.
A: 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.
A: If $S$ is symmetric and $D$ is diagonal then $S+D$ is symmetric, so its eigenvalues are all real, so if your "desired set" contains any nonreal numbers you are bound to be disappointed. 
A: Please read the paper "Honeycombs and sums of Hermitian Matices" by A. Knutson and T. Tao
