CounterexampleConstruction: AI-adaptedVerification: AI-generatedprecheck passaudited 2026-08-31

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

Statement refuted

Moderate ordinary conditioning by itself prevents GMRES stagnation.

The counterexample below shows a nonnormal $2 \times 2$ matrix with $κ_{2} (A) = (3 + \sqrt{5}) / 2$ for which the first GMRES step does not reduce the residual at all.

Facts & Assumptions

Given: The matrix $A = (\begin{array}{cc} 0 & 2 \\ 1 & 1 \end{array})$ , the right-hand side $b = e_{1}$ , and the initial guess $x_{0} = 0$ .

[L1]

The diagonalizable GMRES bound keeps the eigenvector-conditioning factor, not just the ordinary conditioning of $A$ (For a diagonalizable matrix, the GMRES residual bound carries the eigenvector-conditioning factor $κ (V)$ ).

Counterexample

technique · direct

1.1

The initial residual is $r_{0} = e_{1}$ , and the one-step GMRES search space is $x = α e_{1}$ . The corresponding residual is $r (α) = e_{1} - α A e_{1} = e_{1} - α e_{2}$ , so $∥ r (α) ∥_{2}^{2} = 1 + α^{2}$ , minimized at $α = 0$ . Thus the first GMRES step leaves the residual norm equal to $1$ .

algebra

2.1

The matrix is invertible with $\det A = - 2$ , and $A^{T} A = (\begin{array}{cc} 1 & 1 \\ 1 & 5 \end{array})$ has eigenvalues $3 \pm \sqrt{5}$ . Hence the singular values are $\sqrt{3 + \sqrt{5}}$ and $\sqrt{3 - \sqrt{5}}$ , so $κ_{2} (A) = \frac{3 + \sqrt{5}}{2} .$ Therefore exact stagnation can occur even when ordinary conditioning is only moderate. This is consistent with [L1], which warns that nonnormal behavior depends on eigenvector geometry, not eigenvalues or $κ_{2} (A)$ alone.

L1step 1.1algebra∎

Depends on

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

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

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Sources

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