TheoremStatement: Literature-sourcedProof: AI-adaptedprecheck passaudited 2026-08-31

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If $μ$ is not an eigenvalue and one simple eigenvalue is uniquely nearest to $μ$ , shifted inverse iteration converges to its eigendirection

Statement

Let $A$ be diagonalisable with eigenpairs $(λ_{i}, v_{i})$ , let $μ \notin {λ_{1}, \dots, λ_{n}}$ , and suppose one simple eigenvalue $λ_{j}$ satisfies

$∣ λ_{j} - μ ∣ < ∣ λ_{i} - μ ∣ (i \neq j) .$

If the start vector has nonzero $v_{j}$ -component, then shifted inverse iteration with shift $μ$ converges projectively to the eigendirection of $v_{j}$ .

Facts & Assumptions

Given: A diagonalisable matrix $A$ , a shift $μ$ not equal to any eigenvalue, and a start vector with nonzero $v_{j}$ -component.

[L1]

Shifted inverse iteration is power iteration for $(A - μ I)^{- 1}$ (Inverse iteration and shifted inverse iteration).

[L2]

Power iteration converges projectively under a simple strictly dominant eigenvalue and a nonzero component in its eigendirection (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).

Proof

technique · direct

1.1

The eigenvectors of $(A - μ I)^{- 1}$ are the same $v_{i}$ , and the corresponding eigenvalues are $(λ_{i} - μ)^{- 1}$ . The hypothesis $∣ λ_{j} - μ ∣ < ∣ λ_{i} - μ ∣$ means $∣ \frac{1}{λ_{j} - μ} ∣ > ∣ \frac{1}{λ_{i} - μ} ∣ (i \neq j) .$

givenalgebra

2.1

Therefore $(A - μ I)^{- 1}$ has a simple eigenvalue of strictly largest modulus in the eigendirection $v_{j}$ . The start vector has nonzero $v_{j}$ -component by hypothesis, so [L2] applies.

L2step 1.1

3.1

Since [L1] identifies shifted inverse iteration with that power iteration, the iterates converge projectively to the eigendirection of $v_{j}$ .

L1step 2.1∎

Depends on

Used by

Shifted inverse iteration can target a non-dominant eigenvalue by moving it closest to the shift Example

Dependency tree · two levels

5 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Andrew Stuart and Jochen Voss, Matrix Analysis and Algorithms (standard reference, not scraped)

If μμ is not an eigenvalue and one simple eigenvalue is uniquely nearest to μμ, shifted inverse iteration converges to its eigendirection