CounterexampleConstruction: AI-generatedVerification: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-08-31

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Equal dominant moduli can make power iteration alternate or cycle instead of converging

Statement refuted

Refuted claim: power iteration converges whenever a matrix has an eigenvalue of largest modulus.

Let

$A = diag (1, - 1), x_{0} = \frac{1}{\sqrt{2}} [\begin{array}{c} 1 \\ 1 \end{array}] .$

Then the power iteration alternates between two vectors and does not converge.

Facts & Assumptions

Given: The diagonal matrix $A$ and start vector $x_{0}$ from the statement.

[L1]

Power iteration is the recurrence $x_{k + 1} = A x_{k} / ∥ A x_{k} ∥_{2}$ (Power iteration with normalisation and Rayleigh-quotient eigenvalue estimates).

Counterexample

technique · direct

1.1L1algebra

Since $A$ is diagonal with entries $1$ and $- 1$ , $x_{1} = \frac{1}{\sqrt{2}} [\begin{array}{c} 1 \\ - 1 \end{array}], x_{2} = \frac{1}{\sqrt{2}} [\begin{array}{c} 1 \\ 1 \end{array}] = x_{0} .$

2.1step 1.1algebra

Therefore $x_{2 k} = x_{0}$ and $x_{2 k + 1} = x_{1}$ for every $k \geq 0$ . The sequence is periodic of period $2$ , so it does not converge.

3.1step 2.1∎

The matrix has dominant modulus $1$ , but equal dominant moduli prevent convergence here, refuting the claim.

Depends on

Power iteration with normalisation and Rayleigh-quotient eigenvalue estimates

Used by

Nothing in the library uses this result yet.

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