CounterexampleConstruction: AI-generatedVerification: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-01

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A nonsymmetric invertible matrix does not fit the CG orthogonality and minimization theory

Statement refuted

The CG orthogonality and minimization theory applies to every invertible matrix.

Facts & Assumptions

Given: The CG recurrence and the GMRES affine-Krylov residual minimizer.

[F1]

To test the proposed extension outside the Hermitian positive-definite domain, define the attempted algebraic updates $α_{k} = \frac{r_{k}^{T} r_{k}}{p_{k}^{T} A p_{k}}, x_{k + 1} = x_{k} + α_{k} p_{k}, r_{k + 1} = r_{k} - α_{k} A p_{k}, p_{k + 1} = r_{k + 1} + β_{k} p_{k}$ with $p_{0} = r_{0}$ and $β_{k} = (r_{k + 1}^{T} r_{k + 1}) / (r_{k}^{T} r_{k})$ . These repeat the formulas used by CG on its legitimate domain (The conjugate-gradient recurrence); they are not claimed to constitute a run under that definition for nonsymmetric $A$ .

[L1]

GMRES is the residual minimizer over an affine Krylov space for a general matrix (The GMRES iterate as the residual minimizer over an affine Krylov space).

Counterexample

technique · direct calculation

1.1

Take $A = (\begin{array}{cc} 1 & 1 \\ 0 & 1 \end{array}), b = (\binom{0}{1}), x_{0} = (\binom{0}{0}) .$ Then $A$ is invertible but not symmetric. From [F1], $r_{0} = p_{0} = (\binom{0}{1}), α_{0} = \frac{1}{1} = 1, x_{1} = (\binom{0}{1}), r_{1} = b - A x_{1} = (\binom{- 1}{0}) .$ Also $β_{0} = \frac{r_{1}^{T} r_{1}}{r_{0}^{T} r_{0}} = 1, p_{1} = r_{1} + β_{0} p_{0} = (\binom{- 1}{1}) .$

F1algebra

2.1

Now $p_{0}^{T} A p_{1} = (0, 1) (\begin{array}{c} 0 \\ 1 \end{array}) = 1 \neq 0.$ So the search directions fail the basic $A$ -conjugacy identity already at the second direction, and the SPD CG theory cannot be transplanted to this nonsymmetric matrix. The appropriate general method here is the affine-Krylov residual minimizer named in [L1], not CG's Hermitian positive-definite theory.

L1step 1.1algebra∎

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Sources

Richard Barrett et al., Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods (standard reference, not scraped)