Power iteration is an eigenvalue algorithm to find the largest eigenvalue and corresponding eigenvector. This algorithm does not compute a matrix decomposition; therefore, you can use it when Α is a very large sparse matrix.
The power iteration algorithm starts with a vector b 0, which can be an approximation to the dominant eigenvector or a random vector. This iteration describes the method:
b k+1 = A b k / || A b k ||
At every iteration, the vector b k is multiplied by matrix Α and normalized.
The sequence (b k ) does not necessarily converge. A subsequence of (b k ) converges to an eigenvector associated with the dominant eigenvalue under these conditions:
- A has an eigenvalue that is strictly greater in magnitude than its other eigenvalues.
- Starting vector b 0 has a nonzero component in the direction of an eigenvector associated with the dominant eigenvalue.