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Subspace iteration converges to the dominant invariant subspace when a spectral gap separates the wanted and unwanted eigenvalues

Statement

Let $F \in {R, C}$ , let $n \geq 2$ , let $p \in {1, \dots, n - 1}$ , let $A \in M_{n} (F)$ be diagonalisable with eigenvalues ordered by $∣ λ_{1} ∣ \geq \dots \geq ∣ λ_{p} ∣ > ∣ λ_{p + 1} ∣ \geq \dots \geq ∣ λ_{n} ∣$ . Let $Q_{0} \in M_{n \times p} (F)$ have orthonormal columns. Write $A = V diag (Λ_{1}, Λ_{2}) V^{- 1}$ in the ordered eigenbasis and write $V^{- 1} Q_{0} = [\begin{array}{c} C_{1} \\ C_{2} \end{array}]$ with $C_{1} \in M_{p} (F)$ . Assume $C_{1}$ is invertible. Then the column space of $Q_{k}$ converges to that dominant invariant subspace. More precisely, in the ordered eigenbasis it is the graph of $Z_{k} : = Λ_{2}^{k} C_{2} C_{1}^{- 1} Λ_{1}^{- k},$ and $∥ Z_{k} ∥_{2} = O (∣ λ_{p + 1} / λ_{p} ∣^{k}) .$

Facts & Assumptions

Given: The field, dimensions, diagonalisable matrix $A$ , spectral gap, initial orthonormal frame $Q_{0}$ , and valid subspace iteration $(Q_{k})$ from the statement, for which the leading coefficient block $C_{1}$ in the statement is invertible.

[L1]

A diagonalisable matrix admits an eigenbasis (A diagonalisable endomorphism is one admitting a basis of eigenvectors, equivalently a diagonal matrix representation).

[L2]

Subspace iteration is the repeated QR orthonormalisation of $A Q_{k}$ (Subspace iteration and the dominant invariant subspace of a matrix).

Proof

technique · direct

1.1L1given

By [L1], use the decomposition from the statement, where $Λ_{1} = diag (λ_{1}, \dots, λ_{p})$ and $Λ_{2} = diag (λ_{p + 1}, \dots, λ_{n})$ . The hypothesis states exactly that $C_{1}$ is invertible.

2.1L2step 1.1algebra

The spectral gap implies $λ_{p} \neq 0$ , so $Λ_{1}^{k} C_{1}$ is invertible for every $k$ . Hence $A^{k} Q_{0}$ has full column rank, every reduced QR step in [L2] is defined, and the column space of $Q_{k}$ equals that of $A^{k} Q_{0}$ . Now $A^{k} Q_{0} = V [\begin{array}{c} Λ_{1}^{k} C_{1} \\ Λ_{2}^{k} C_{2} \end{array}] .$ Multiplying on the right by $C_{1}^{- 1} Λ_{1}^{- k}$ shows that the same column space is the graph of $Λ_{2}^{k} C_{2} C_{1}^{- 1} Λ_{1}^{- k}$ over the dominant invariant subspace.

3.1step 2.1algebra

The spectral gap implies $∥ Λ_{2}^{k} C_{2} C_{1}^{- 1} Λ_{1}^{- k} ∥_{2} = O ⁣ ({∣ \frac{λ_{p + 1}}{λ_{p}} ∣}^{k}) .$ Hence the graph in step 2.1 converges to the dominant invariant subspace at that rate.

4.1step 3.1∎

Therefore the column space of $Q_{k}$ converges to the dominant invariant subspace, with the precise graph-norm rate stated above.

Depends on

Used by

Under separated moduli and leading-minor hypotheses, unshifted QR drives the strict lower triangle to zero and orders the eigenvalues on the diagonal Theorem

Dependency tree · two levels

7 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)