Iterate . For a self-adjoint operator with a unique dominant eigenvalue separated in modulus from the others, a starting vector that is nonorthogonal to its dominant eigenvector gives convergence of the eigendirection at the spectral-ratio rate. For a general diagonalizable matrix, the condition is a nonzero coefficient in the dominant component of its eigenbasis, detected by the corresponding left eigenvector; an ordinary dot product with the right eigenvector does not generally detect that coefficient.
The power method is power iteration; its convergence depends on the dominant spectral component of the starting vector.
For example, has dominant right eigenvector and a dominant left eigenvector . The starting vector has nonzero dot product with the right eigenvector, but : it stays forever in the eigenspace for the smaller eigenvalue . Its coefficient along the dominant direction is zero, as the left eigenvector detects. An orthonormal eigenbasis makes the two tests agree in the self-adjoint case.
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