Suppose $A$ is a real symmetric matrix and its eigenvalue decomposition $V \Lambda V^T$ is given. It is easy to see what happens with the eigenvalues of the sum $A + cI$ where $c$ is a scalar constant (see this question). Can we draw any conclusion in the general case $A + D$ where $D$ is an arbitrary diagonal matrix? Thanks.Regards,Ivan
Eigenvalue decomposition of the sum: A (symmetric) + D (diagonal)
linear algebra
One can say very little, except for generalities such as that the eigenvalues change continuously with the entries of $D$. You can see by symbolic computation in the 2 by 2 case that nothing strong can be expected.