DefinitionDefinition: Literature-sourcedProof: Not applicableaudited 2026-08-31

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Subspace iteration and the dominant invariant subspace of a matrix

Definition

Let $A \in M_{n} (F)$ , let $p \in {1, \dots, n}$ , and let $Q_{0} \in M_{n \times p}$ have orthonormal columns. The subspace iteration is defined by the reduced QR recurrences

$Z_{k} : = A Q_{k}, Z_{k} = Q_{k + 1} R_{k + 1},$

where $Q_{k + 1}$ has orthonormal columns and $R_{k + 1}$ is upper triangular.

If $p < n$ and $A$ is diagonalisable with eigenvalues ordered so that $∣ λ_{1} ∣ \geq \dots \geq ∣ λ_{n} ∣$ and $∣ λ_{p} ∣ > ∣ λ_{p + 1} ∣$ , the span of the first $p$ eigenvectors is the dominant invariant subspace of dimension $p$ . For $p = n$ , the dominant invariant subspace is the whole space $F^{n}$ and no spectral-gap condition is needed.

Depends on

Used by

Dependency tree · two levels

8 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)