Techniques for lower-bounding angle between two eigenvectors of a matrix

Question

Are there any techniques for lower-bounding the angle between eigenvectors of a matrix? Or a lower bound on the related quantity of the condition number of the matrix of eigenvectors? In particular I'm looking for bounds that depend on the difference in the corresponding eigenvalues, with larger angles when the eigenvalues are more separated.

For symmetric matrices (and more generally normal matrices) the angles are of course all right angles. I'm looking for techniques that apply to non-normal matrices.

(The particular class of matrices that I care about is stochastic matrices with trace $n-1$ as described in my previous question Bounds on $||P^{k+1} - P^k||$ for $n$ by $n$ stochastic matrix $P$ with trace $n-1$ and integer $k>>n$. .)

Answer 1

Here's a generalization of the result that eigenvectors of a real symmetric matrix with distinct eigenvalues are orthogonal:

Let $x$ and $y$ be unit eigenvalues of a real matrix $A$ with eigenvalues $\lambda$ and $\mu$, respectively. Then $$ \lambda\langle x,y\rangle =\langle Ax,y\rangle =\langle x,A^\mathrm{T}y\rangle =\langle x,Ay\rangle-\langle x,(A-A^\mathrm{T})y\rangle =\overline{\mu}\langle x,y\rangle-\langle x,(A-A^\mathrm{T})y\rangle. $$ Rearranging and applying standard inequalities then yields $$ |\langle x,y\rangle| \leq\frac{\|A-A^\mathrm{T}\|_2}{|\lambda-\overline{\mu}|}. $$ The numerator of this fraction makes intuitive sense, since any correlation between eigenvectors must come from the skew-symmetric part of the matrix. Furthermore, the denominator provides the relationship you suggested in your question, especially when the eigenvalues are real.