TITLE: Eigenvalues of matrix sums
QUESTION [75 upvotes]: Is there a relationship between the eigenvalues of individual matrices and the eigenvalues of their sum? What about the special case when the matrices are Hermitian and positive definite?
I am investigating this with regard to finding the normalized graph cut under general convex constraints. Any pointers will be very helpful.

REPLY [4 votes]: Here is a trivial case with a simple solution. Applicable in Quantum Mechanics, for one.
Given two matrices of the form $A \otimes Id$, $Id \otimes B$, the eigenvalues of their sum are all combinations $a_i+b_j$, where $A\vec{a}_i=a_i\vec{a}_i$ and $B\vec{b}_i=b_i\vec{b}_i$. The eigenvectors are all tensor products of the individual eigenvectors of $A$ and $B$.
This is a one-liner to show:
$$(A \otimes Id + Id \otimes B) \,\vec{a}_i\otimes \vec{b}_j=(a_i+b_j)\,\vec{a}_i\otimes \vec{b}_j$$