# Eigenvalue decomposition of a symmetric matrix minus a diagonal matrix

I have a symmetric matrix $\Sigma$ and a diagonal matrix $D$. I know that eigenvalue decomposition of $\Sigma = Q\Lambda Q^T$. Now I would like to find the eigenvalues and the eigenvectors of the following matrix: $$\Sigma_v = \Sigma-D$$ I understand that if $D=nI$, things are trivial. But in a general case, what is the best approach to understand how much the eigenvalues and vector change from that of $\Sigma$, due to the diagonal perturbation $D$?

• Hi Sai. You did well to format your question using MathJax, but please take note of the formatting changes I've made. Most of the time, you'll get a better result if you put the entire mathematical expression in $...$ – Omnomnomnom Jan 14 '17 at 16:50
• Thank you! I will keep this in mind next time. – Sai Ganesh Jan 14 '17 at 16:58

Unfortunately, there isn't really anything special about the case where $D$ is diagonal (as opposed to an arbitrary symmetric perturbation). In particular, for any $P = UDU^T$, the matrix $\Sigma - P$ is similar to $U^T \Sigma U - D$.
• Are you particularly interested in the $2 \times 2$ case? There's a lot you can get away with then. I assume that with "different ellipsoids", you're referring to the image of the unit circle under each transformation. Sure, that might lead you to something interesting. Since all the characterstic polynomials are degree $2$, you might actually be able to come up with an analytic expression for eigenvalues/eigenvectors as a function of the diagonal entries. – Omnomnomnom Jan 14 '17 at 17:08