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Consider the symmetric matrix and initial unit vectorIts orthonormal eigenbasis consists of $u_1=(1,1{)}^T/\sqrt{2}$ and $u_2=(1,-1{)}^T/\sqrt{2}$, with eigenvalues $3$ and $1$. Since $x_0=(u_1+u_2)/\sqrt{2}$, direct multiplication givesThe first unnormalized vectors are $(1,0{)}^T$, $(2,1{)}^T$, $(5,4{)}^T$, and $(14,13{)}^T$. If ${\theta }_k$ is the acute angle from the line spanned by the dominant eigenvector, then $\tan {\theta }_k=3^{-k}$. Every application divides the unwanted-to-wanted component ratio by three: normalization changes lengths, while the spectral gap selects the direction.

The right panel compares the power iteration direction error, Rayleigh quotient error, and eigenpair residual. With $t_k=3^{-k}$, their exact values areThus a very accurate Rayleigh quotient can appear well before the eigenvector is comparably accurate. This is an example of quadratic Rayleigh-quotient improvement for the power method.

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 ${\lambda }_1=|{\lambda }_1|e^{i\varphi }$, where $\varphi$ is its complex argument. The leading factor turns by $e^{i\varphi }$ per step, so the vectors generally do not converge. For example, $A=\mathrm{diag}(2i,1)$ and $x_0=(1,1{)}^T/\sqrt{2}$ give $A^k{x}_0=((2i{)}^k,1{)}^T/\sqrt{2}$: 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 $e^{-i(\arg c_1+k\varphi )}$ gives a sequence converging to $u_1$.

A stopping rule based only on $\Vert x_{k+1}-x_k{\Vert }_2$ therefore rejects successful sign-alternating iterations. A comparison invariant under complex unit factors isEven this comparison can be small for a slowly changing wrong mixture; the eigenpair residual supplies a separate diagnostic.