To compute the eigenvector corresponding to dominant eigenvalue of a symmetric matrix $A\in\mathbb{R}^{n\times n}$, one used Power Iteration, i.e., given some random initialization, $u_1\in\mathbb{R}^n$, one iteratively computes $$u_1\leftarrow Au_1,$$after which a normalization is applied to $u_1$. Now, suppose that eigenvectors $u_1, u_2$ are computed in advance, and one wants to compute the eigenvector $u_3$ associated with the third dominant eigenvalue.
In case the initial $u_3$ is orthogonal to both $u_1$ and $u_2$, can it be shown that the series $$u_3\leftarrow Au_3$$ converges in the direction of the eigenvector of $A$ corresponding to third dominant eigenvalue.
Note that the question is motivated by an observation that most implementations of the Power Iteration that computes higher eigenvectors provides Gram-Schmidt (GS) orthonormalization in each iteration, i.e., after each $u_3\leftarrow Au_3$, orthogonalization is applied wrt $v_1, v_2$. In case the orthonogonality of $u_3$ is imposed with its initialization (ie. from the start), is the GS necessary after each matrix-vector multiplication?