ExampleConstruction: AI-generatedVerification: AI-generatedprecheck passaudited 2026-08-31

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Power iteration on a diagonal matrix shows the $∣ λ_{2} / λ_{1} ∣^{k}$ convergence rate explicitly

Example

Let

$A = diag (4, 1), x_{0} = \frac{1}{\sqrt{2}} [\begin{array}{c} 1 \\ 1 \end{array}] .$

Then

$x_{k} = \frac{1}{\sqrt{1 + 16^{- k}}} [\begin{array}{c} 1 \\ 4^{- k} \end{array}],$

so the error in direction is exactly of order $4^{- k} = ∣ λ_{2} / λ_{1} ∣^{k}$ .

Facts & Assumptions

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

[L1]

For an $n \times n$ diagonalisable matrix with $n \geq 2$ , a simple strictly dominant eigenvalue, and a start vector having nonzero component in its eigendirection, power iteration converges projectively at the dominant eigenvalue ratio (If a diagonalisable matrix has a simple eigenvalue of strictly largest modulus and the start vector has a nonzero component in that eigendirection, power iteration converges projectively at the eigenvalue-ratio rate).

Verification

technique · computation

1.1

Since $A^{k} x_{0} = 2^{- 1 / 2} (4^{k}, 1)^{T}$ , normalising gives $x_{k} = \frac{1}{\sqrt{16^{k} + 1}} [\begin{array}{c} 4^{k} \\ 1 \end{array}] = \frac{1}{\sqrt{1 + 16^{- k}}} [\begin{array}{c} 1 \\ 4^{- k} \end{array}] .$

algebra

2.1

The second coordinate of $x_{k}$ is asymptotic to $4^{- k}$ , so the angle to the dominant eigendirection $F e_{1}$ decays like $4^{- k}$ . This is exactly the rate predicted by [L1].

L1step 1.1algebra∎

Depends on

If a diagonalisable matrix has a simple eigenvalue of strictly largest modulus and the start vector has a nonzero component in that eigendirection, power iteration converges projectively at the eigenvalue-ratio rate

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

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Power iteration on a diagonal matrix shows the ∣λ2/λ1∣k∣λ2​/λ1​∣k convergence rate explicitly