For the Power Method I make an initial guess for the dominant eigenvector of matrix A. It can be expressed as a linear combination of the set of eigenvectors of A.
Why is it that the Power Method converges to the dominant eigenvector? Is it that the initial guess contains a non-zero component along the dominant eigenvector (by assumption) and since it has the largest (by magnitude) eigenvalue of A the scaling within the recurrence relation "brings" us closer and closer to the dominant eigenvector while the non-dominant eigenvectors are being "silenced" by the scaling?