TITLE: Eigenvalues, singular values, and the angles between eigenvectors
QUESTION [5 upvotes]: Suppose the $n \times n$ matrix $A$ has eigenvalues $\lambda_1, \ldots, \lambda_n$ and singular values $\sigma_1, \ldots, \sigma_n$. It seems plausible that by comparing the singular values and eigenvalues we gets some sort of information about eigenvectors. Consider:
a. The singular values are equal to the absolute values of eigenvalues if and only if the matrix is normal, i.e., the eigenvectors are orthogonal (see http://en.wikipedia.org/wiki/Normal_matrix , item 11 of the "Equivalent definitions" section ).
b. Suppose we have two distinct eigenvalues $\lambda_1, \lambda_2$ with eigenvectors $v_1, v_2$. Suppose, hypothetically, we let $v_1$ approach $v_2$, while keeping all the other eigenvalues and eigenvectors the same. Then the largest singular value approaches infinity. This follows since $\sigma_{\rm max} = ||A||_2$ and $A$ maps the vector $v_1 - v_2$, which approaches $0$, to $\lambda_1 v_1 - \lambda_2 v_2$, which does not approach $0$.
It seems reasonable to guess that the ``more equal'' $|\lambda_1|, \ldots, |\lambda_n|$ and $\sigma_1, \ldots, \sigma_n$ are, the more the eigenvectors look like an orthogonal collection.  So naturally my question is whether there is a formal statement to this effect.

REPLY [2 votes]: A little help in understanding the relation or difference between eigenvectors and singular vectors is elaborated in detail in: SVD vs EVD

The bookmark leads you to a comparison between evd and svd. But there's much more there if you browse through the pages.
EngleField and Farr show an elegant derivation for the angle between eigenvectors:
Englefield, M. J., & Farr, G. E. (2006). Eigencircles of 2 x 2 Matrices. Mathematics Magazine Vol. 79 Oct.,2006, 281-289. on page 285
The angle between the singular vectors is the angle of the rotation $UV^T$ where $U$ and $V$ contain the singular vectors of $ A = U \Sigma V^T$
Eigencircle representation leads you to a figure showing this angle on an eigencircle plot.