#eigenvectors

Question 1: Single choice

The power method iterates $\mathbf{v}_{k+1} = \Sigma \mathbf{w}_k$ , $\mathbf{w}_{k+1} = \mathbf{v}_{k+1}/\lVert \mathbf{v}_{k+1} \rVert_2$ for a symmetric positive definite $\Sigma$ (e.g. a data covariance matrix) with eigenvalues $\lambda_1 > \lambda_2 > \cdots > \lambda_n > 0$ and orthonormal eigenvectors $\mathbf{u}_i$ . Starting from $\mathbf{w}_0$ with $\mathbf{u}_1^\top \mathbf{w}_0 \neq 0$ , what happens?

$\mathbf{w}_k$ converges to $\mathbf{u}_n$ , the eigenvector of the smallest eigenvalue1
It converges to $\pm\mathbf{u}_1$ ; the other components shrink like $(\lambda_i/\lambda_1)^k$ 2
$\mathbf{w}_k$ converges to $\mathbf{u}_1$ at a speed that does not depend on the eigenvalues3
$\mathbf{w}_k$ reaches $\mathbf{u}_1$ exactly after $n$ iterations4

Timed test