I have a question that might be answered with a pointer to some references or with some discussion. I did some searching, to no avail, but I realized that I might not have the vocabulary to form a successful search.


Suppose we have a random (or pseudorandom) vector $r\in\mathbb{R}^{n}$ with $\left\|r\right\|=1$, e.g. generated by the rand() function in Matlab and normalized. Let $S\in\mathbb{R}^{n\times n}$ be a symmetric matrix with the orthogonal decomposition $S=V^{T}DV$ where the columns of $V$ are orthonormal and $D$ is the diagonal matrix of eigenvalues. Let $v\_{i}$ be a column of $V$. What is the probability that $\left|r^{T}v\_{i}\right|< \alpha$ where $\alpha\in\left(0,1\right)$. In other words, what is the probability that $r$ has only a certain size component in a particular eigenvector?



On Random Vectors and Eigenvectors of Symmetric Matrices
As pointed out in the comments, the matrix has nothing to do with the question, and you are simply trying to compute the distribution of $x \cdot v$ as $x$ varies over the unit sphere, and $v$ is a fixed unit vector. By rotational invariance, you might as well assume that $v = (1, 0, \dots, 0),$ at which point the question becomes an easy integration exercise.


**Edit** if you don't want to bother integrating, the concentration of measure phenomenon (first observed by Boltzmann, I believe) is that the distribution of the areas of cross sections of the sphere is essentially normal for moderate $n.$ ( you can easily compute the variance), so then you can approximate the answer to your question by an inverse error function.


