PropositionStatement: AI-adaptedProof: AI-generatedprecheck passaudited 2026-08-31

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For a diagonalizable matrix, the GMRES residual bound carries the eigenvector-conditioning factor $κ (V)$

Statement

Assume $A = V Λ V^{- 1}$ is diagonalizable, with $Λ = diag (λ_{1}, \dots, λ_{n})$ . If $r_{m}$ is the $m$ th GMRES residual, then

$∥ r_{m} ∥_{2} \leq κ_{2} (V) \min_{\begin{array}{c} \deg p \leq m \\ p (0) = 1 \end{array}} \max_{1 \leq i \leq n} ∣ p (λ_{i}) ∣ ∥ r_{0} ∥_{2},$

where $κ_{2} (V) = ∥ V ∥_{2} ∥ V^{- 1} ∥_{2}$ .

Facts & Assumptions

Given: A diagonalization $A = V Λ V^{- 1}$ and an $m$ th GMRES residual $r_{m}$ .

[L1]

One has $r_{m} = p_{m} (A) r_{0}$ for some polynomial $p_{m}$ with $\deg p_{m} \leq m$ and $p_{m} (0) = 1$ , and GMRES chooses such a polynomial minimizing the residual norm (The GMRES residual has the form $r_{m} = p_{m} (A) r_{0}$ with $p_{m} (0) = 1$ and $\deg p_{m} \leq m$ ).

Proof

technique · direct

1.1L1algebra

By [L1], $r_{m} = p_{m} (A) r_{0} = V p_{m} (Λ) V^{- 1} r_{0} .$ Hence $∥ r_{m} ∥_{2} \leq ∥ V ∥_{2} ∥ p_{m} (Λ) ∥_{2} ∥ V^{- 1} ∥_{2} ∥ r_{0} ∥_{2} .$ Since $p_{m} (Λ)$ is diagonal, its operator norm is $\max_{i} ∣ p_{m} (λ_{i}) ∣$ . Therefore $∥ r_{m} ∥_{2} \leq κ_{2} (V) \max_{i} ∣ p_{m} (λ_{i}) ∣ ∥ r_{0} ∥_{2} .$

2.1L1step 1.1∎

The polynomial $p_{m}$ from [L1] minimizes $∥ p (A) r_{0} ∥_{2}$ over all polynomials with $\deg p \leq m$ and $p (0) = 1$ , so the bound from step 1.1 also holds after taking the minimum over that admissible class.

Depends on

The GMRES residual has the form $r_m=p_m(A)r_0$ with $p_m(0)=1$ and $\deg p_m\le m$

Used by

A nonnormal matrix can make GMRES stagnate even when ordinary conditioning is not extreme Counterexample

Dependency tree · two levels

3 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

Lloyd N. Trefethen and David Bau III, Numerical Linear Algebra (standard reference, not scraped)

For a diagonalizable matrix, the GMRES residual bound carries the eigenvector-conditioning factor κ(V)κ(V)