= Power iteration direction error
{title2=$\varepsilon_k=\|(I-uu^*)x_k\|_2$}
For a <unit vector> $u$ spanning the target <eigenspace> and a normalized iterate $x_k$, define
$$
\varepsilon_k=\|(I-uu^*)x_k\|_2=\sqrt{1-|u^*x_k|^2}.
$$
This is the <norm> of the component in the <orthogonal complement> of $u$, equivalently the sine of the acute <angle> to its line. It ignores the arbitrary sign or complex unit factor of an <eigenvector>. Under <power iteration convergence for a diagonalizable matrix>, $\varepsilon_k=O(\gamma^k)$ in <Big O notation>. The <orthogonal projection> defines how we measure direction; it does not assume that the <eigenbasis> of $A$ is <orthogonal>. In computation, the projection form avoids subtracting nearly equal numbers inside the square root.
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