= Sign and phase in power iteration
{title2=$x_k=\frac{c_1\lambda_1^k}{|c_1\lambda_1^k|}u_1+O(\gamma^k)$}
Under <power iteration convergence for a diagonalizable matrix>, take $v_1=u_1$ to be a <unit vector>. The displayed formula describes both the dominant <eigenvector> direction and its rotating complex unit factor. With real data and a positive <dominant eigenvalue>, $x_k$ converges to $u_1$ or $-u_1$ according to the sign of $c_1$. With a negative <dominant eigenvalue>, consecutive <vectors> approach opposite signs even though their <power iteration direction error> tends to zero.
For complex data write $\lambda_1=|\lambda_1|e^{i\phi}$, where $\phi$ is its <complex argument>. The leading factor turns by $e^{i\phi}$ per step, so the <vectors> generally do not converge. For example, $A=\operatorname{diag}(2i,1)$ and $x_0=(1,1)^T/\sqrt2$ give $A^kx_0=((2i)^k,1)^T/\sqrt2$: the first component cycles through four phases while the ratio of component <moduli> tends to zero. Its <power iteration direction error> is $2^{-k}/\sqrt{1+2^{-2k}}$. Multiplying $x_k$ by $e^{-i(\arg c_1+k\phi)}$ gives a sequence converging to $u_1$.
A stopping rule based only on $\|x_{k+1}-x_k\|_2$ therefore rejects successful sign-alternating iterations. A comparison invariant under complex unit factors is
$$
\min_{|\omega|=1}\|x_{k+1}-\omega x_k\|_2=\sqrt{2-2|x_k^*x_{k+1}|}.
$$
Even this comparison can be small for a slowly changing wrong mixture; the <eigenpair residual> supplies a separate diagnostic.
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