= Power iteration failure cases
A tied largest <modulus> can prevent even direction convergence. With $D=\operatorname{diag}(2,-2)$ and $x_0=(1,1)^T/\sqrt2$,
$$
x_k=\frac1{\sqrt2}\begin{pmatrix}1\\(-1)^k\end{pmatrix},\qquad \rho_k=0,\qquad \|r_k\|_2=2.
$$
The two directions form a cycle; neither is an <eigenvector>. There is no dominant-<modulus> <spectral gap>. A tied <modulus> is not a universal failure: an initial <eigenvector> remains on its line, and $A=2I$ preserves every direction. <Subspace iteration> addresses a separated cluster by targeting a <dominant invariant subspace> instead of one <eigenvector>.
An <orthogonal> start can fail despite a strict <spectral gap>. For $A=\begin{pmatrix}2&1\\1&2\end{pmatrix}$ and $x_0=(1,-1)^T/\sqrt2$, the dominant component is zero, so $Ax_0=x_0$ and every iterate equals $x_0$. The <Rayleigh quotient> is exactly $1$ and the <eigenpair residual> is zero, although the <dominant eigenvalue> is $3$. Exact arithmetic never creates a missing component.<Templates for the Solution of Algebraic Eigenvalue Problems: A Practical Guide/Power Method>{ref}
\Image[/power-iteration-tied-modulus-orthogonal-start.png]
{title=Tied <eigenvalue> <moduli>, an <orthogonal> start, and sign alternation in <power iteration>}
The third panel contrasts these failures with $N=\operatorname{diag}(-3,1)$, $x_0=(1,1)^T/\sqrt2$, for which
$$
x_k=\frac1{\sqrt{1+3^{-2k}}}\begin{pmatrix}(-1)^k\\3^{-k}\end{pmatrix}.
$$
Here the horizontal <eigenvector> direction converges with a strict <spectral gap>, while the normalized <vectors> alternate signs. This is the successful direction convergence described under <sign and phase in power iteration>.
For a <nonorthogonal> <eigenbasis>, even the word “orthogonal” must be used carefully. The <diagonalizable matrix> $C=\begin{pmatrix}3&2\\0&1\end{pmatrix}$ has right <eigenvectors> $v_1=(1,0)^T$, $v_2=(-1,1)^T$, and dominant <left eigenvector> $w_1=(1,1)^T$. For $x_0=(a,b)^T$, the dominant coefficient is $c_1=w_1^Tx_0=a+b$. The starting <vector> $v_2$ is <nonorthogonal> to $v_1$ but stays at the smaller <eigenvalue> $1$; $(0,1)^T$ is <orthogonal> to $v_1$ but has $c_1=1$ and converges to the dominant direction. The <left eigenvector> test, not the right-<eigenvector> <inner product>, controls the outcome.
A <random vector> drawn from a <continuous probability distribution> with a <probability density function> on the full real or complex <vector space> has $c_1\ne0$ with probability one. This avoids the exact missing-component <hyperplane>, but does not guarantee a large component or fast convergence. <Floating-point arithmetic> may seed a missing component, yet a structurally preserved zero can remain zero, so accidental perturbations are not a convergence guarantee.
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