= Power method
{title2=$q^{(k+1)}=Aq^{(k)}/\|Aq^{(k)}\|$}
Iterate $q^{(k+1)}=Aq^{(k)}/\|Aq^{(k)}\|$. 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, $B=\begin{pmatrix}2&1\\0&1\end{pmatrix}$ has dominant right <eigenvector> $(1,0)^T$ and a dominant <left eigenvector> $(1,1)^T$. The starting <vector> $q^{(0)}=(-1,1)^T/\sqrt2$ has nonzero <dot product> with the right <eigenvector>, but $Bq^{(0)}=q^{(0)}$: it stays forever in the <eigenspace> for the smaller <eigenvalue> $1$. 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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