rank-one perturbation of a matrix corresponding to a specific spectrum Let $A$ be a real symmetric matrix whose spectrum is $\lambda_1,\lambda_2,\ldots,\lambda_n$.
Let $A'$ be the matrix obtained by adding a perturbation to $A$. The requirement is that only the second eigenvector of $A'$ is known and fixed in advance. Is there a way to obtain the set of all possible perturbations which will guarantee that $A'$ has the specified second eigenvector?
 A: The perturbation matrix $\Pi$  must have $v_2$ (your second eigenvector) as an eigenvector. The associated eigenvalue is either $0,$ or not. In the second case, $\Pi = c v_2 v_2^t$ (in other words, a multiple of the projection onto the space spanned by $v_2.$ In the first case, $\Pi = c w w^t,$ where $(w, v_2) = 0$ (in English, a multiple of a projection onto some one-dimensional subspace orthogonal to $v_2.$ As for the ordering of the eigenvalues (as mentioned by @Denis), this is very easy to check in the first case ($c < \lambda_1 - \lambda_2$), and slightly less easy in the second case (you need to compute the projection of $w$ onto $v_1$).
A: @Pietro Majer,
A few clarifications in the question.
I am interested in the case of the matrix being the Laplacian of some undirected weighted graph. Thus, A would now be real,symmetric and positive semi definite. 
Also A' = A + \epsilon H where H is rank-one. Importantly the second eigen-vector i meant was the one the corresponds to the second "smallest" eigen-value.  
Thanks. 
