Projected eigenvalue problem: how do eigenvectors change when subspace is slightly changed?

Question

Let $A \in \mathbb{R}^{n \times n}$ and let $V_1 = [v_1, v_2] \in \mathbb{R}^ {n \times 2} $ en $V_2 = [v_1, \hat{v}_2]\in \mathbb{R}^ {n \times 2} $ two orthonormal matrices with $v_2 \approx \hat{v}_2$. How is the maximal eigenvalue $\lambda_1^{(2)}$ and eigenvector $x^{(2)}_1$ of $$V_2^T A V_2 = \begin{bmatrix} v_1^T A v_1 & v_1^T A \hat{v}_2 \\ \hat{v}_2^T A v_1 & \hat{v}_2^T A \hat{v}_2 \end{bmatrix}$$ relate to the maximal eigenvalue $\lambda^{(1)}_1$ and eigenvector $x^{(2)}_1$ of $$V_1^T A V_1 = \begin{bmatrix} v_1^T A v_1 & v_1^T A v_2 \\ v_2^T A v_1 & v_2^T A v_2 \end{bmatrix}.$$ I guess you can use perturbation analysis on this since $$ V_2^T A V_2 = V_1^T A V_1 + \begin{bmatrix} 0 & v_1^T A (\hat{v}_2 - v_2) \\ (\hat{v}_2 - v_2)^T A v_1 & \hat{v}_2^T A \hat{v}_2 - v_2^T A v_2 \end{bmatrix} $$ but I cannot find a theorem for this. I would guess you can relate the angle between the eigenvectors to the angle betweeen $v_2$ and $\hat{v}_2$. Can anyone help me?