Given a matrix M and a vector v, is there an efficient method to find the normalized eigenvector of M that is closest to v, in that it has maximal overlap.
In particular I'm interested in Hermitian matrices, often symmetric and often sparse. By efficient I mean that I should be able to find $R<N$ eigenvectors of an $N \times N$ matrix with the largest overlap with better scaling than diagonalising the full matrix then sorting.
I think this is an interesting question because all algorithms (at least that I know) work by utilising the properties of the dominant eigenvalue - that multiple applications of the matrix projects onto the eigenvector with that eigenvalue. By shifting/inverting it is possible to pick which should be the dominant eigenvalues. It therefore seems that these algorithms fundamentally can't pick out specific eigenvectors only eigenvalues. I am wondering if it is even possible to select a particular eigenvector.
Note that this question has been asked (over three years ago) but with no answer. Find the eigenvector with maximum overlap
Please see https://scicomp.stackexchange.com/questions/28111/eigenvector-with-maximum-overlap/30528#30528 for the solution to this problem.
