Let $A$ be an Hermitian $n \times n$ matrix and $w$ be a fixed vector $\in \mathbb{R}^n$ of norm one, i.e. $\|w\| = 1$. Is there a way to find the set of eigenvectors of $A$ that have the greatest scalar product with $w$ among all eigenvectors of $A$?
I.e. is there an iterative method to solve the following problem $$ \text{argmax}_{Q, \lambda} \ \ \|Qw\|^2 \text{ s.t. } QA = \left(\begin{array}{ccc} \lambda_1 & & \\ & \ddots & \\ & & \lambda_k \end{array}\right) Q \text{ and } QQ^T = I$$ where $Q$ is a $k \times n$ matrix that collects $k$ eigenvectors, one per row.