I know that the above doesn't exist, but do bear with me. I need estimates/formulas for entries of certain eigenvectors and Cramer's rule keeps popping up in my mind. So, what can play an anlogous role in this case?



Cramer's rule for eigenvectors
If $u$ is a unit eigenvector with eigenvalue $\lambda$ of a Hermitian matrix
$$ A\_n = \begin{pmatrix} a & X^\* \\\\ X & A\_{n-1} \end{pmatrix}$$
with $a$ a real number, $X$ an $n-1$-dimensional row vector, and $A\_{n-1}$ an $n-1 \times n-1$ Hermitian matrix, then (provided that $\lambda$ is not an eigenvalue of $A\_{n-1}$) the magnitude of the first coefficient $u\_1$ of $u$ is given by the formula 


$$ |u\_1|^2 = \frac{1}{1 + \| (A\_{n-1} - \lambda)^{-1} X \|^2};$$


see e.g. Lemma 41 of [this paper of mine](http://arxiv.org/abs/0906.0510) with Van Vu (and we learned of this formula from [this previous paper](http://arxiv.org/abs/0711.1730) of Erdos, Schlein, and Yau). Similarly for other components of $u$. The proof proceeds by expanding out the bottom $n-1$ components of the eigenvalue equation $A\_n u = \lambda u$ and using the resulting equation and the formula $\|u\|^2=1$ to solve for $u\_1$. 


The expression $\| (A\_{n-1} - \lambda)^{-1} X \|$ can be expanded further by a number of formulae (e.g. Cramer's rule); in our applications, it turns out that the spectral theorem applied to $A\_{n-1}$ is useful. The above identity is particularly useful for establishing delocalisation of eigenvectors for random matrices (i.e. that the energy of the coefficients of a unit vector are spread out almost uniformly). It may seem a bit strange that the top left coefficient $a$ plays no explicit role in this formula, but it is implicitly present due to its influence on the eigenvalue $\lambda$.


The phase $u\_1$ of the unit eigenvector does not have a clean formula, because eigenvectors are only determined up to phase rotations unless one somehow selects an artificial normalisation.


