first column of is an eigenvector of corresponding to the dominant eigenvalue . Since the dominant eigenvalue of is unique, the first Jordan block of is the matrix where is the largest eigenvalue of A in magnitude. The starting vector can be written as a linear combination of the columns of V: By assumption, has a nonzero component in the direction of the dominant eigenvalue, so . The computationally useful recurrence relation for can be rewritten as: where the expression: is more amenable to the following analysis. The expression above simplifies as The limit follows from the fact that the eigenvalue of is less than 1 in magnitude, so It follows that: Using this fact, can be written in a form that emphasizes its relationship with when k is large: where and as The sequence is bounded, so it contains a convergent subsequence. Note that the eigenvector corresponding to the dominant eigenvalue is only unique up to a scalar, so although the sequence may not converge, is nearly an eigenvector of A for large k. Alternatively, if A is diagonalizable, then the following proof yields the same result Let λ1, λ2, ..., λm be the m eigenvalues (counted with multiplicity) of A and let v1, v2, ..., vm be the corresponding eigenvectors. Suppose that is the dominant eigenvalue, so that for . The initial vector can be written: If is chosen randomly (with uniform probability), then c1 ≠ 0 with probability 1. Now, On the other hand: Therefore, converges to (a multiple of) the eigenvector . The convergence is geometric, with ratio where denotes the second dominant eigenvalue. Thus, the method converges slowly if there is an eigenvalue close in magnitude to the dominant eigenvalue. Applications Although the power iteration method approximates only one eigenvalue of a matrix, it remains useful for certain computational problems. For instance, Google uses it to calculate the PageRank of documents in their search engine, and Twitter uses it to show users recommendations of whom to follow. The power iteration method is especially suitable for sparse matrices, such as the web matrix, or as the matrix-free method that does not require storing the coefficient matrix explicitly, but can instead access a function evaluating matrix-vector products . For non-symmetric matrices that are well-conditioned the power iteration method can outperform more complex Arnoldi iteration. For symmetric matrices, the power iteration method is rarely used, since its convergence