PropositionStatement: AI-adaptedProof: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-01

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Galerkin orthogonality turns residual orthogonality into $A$ -orthogonality of the error, and the Galerkin iterate is unique

Statement

Let $A$ be Hermitian positive definite, let $A x = b$ , let $x_{0}$ be an initial guess, let $r_{0} = b - A x_{0}$ , let $x_{*}$ be the exact solution of $A x = b$ , and let $x_{m} \in x_{0} + K_{m} (A, r_{0})$ . Then the following are equivalent:

$b - A x_{m}$ is orthogonal to $K_{m} (A, r_{0})$ in the standard inner product.
$x_{*} - x_{m}$ is orthogonal to $K_{m} (A, r_{0})$ in the energy inner product.

When these conditions hold, the vector $x_{m}$ is the unique element of $x_{0} + K_{m} (A, r_{0})$ with that property.

Facts & Assumptions

Given: A Hermitian positive-definite system $A x = b$ , an initial guess $x_{0}$ , $r_{0} = b - A x_{0}$ , the exact solution $x_{*}$ , and a vector $x_{m} \in x_{0} + K_{m} (A, r_{0})$ .

[F1]

A Krylov Galerkin iterate is an element of $x_{0} + K_{m} (A, r_{0})$ whose residual is orthogonal to $K_{m} (A, r_{0})$ (The $m$ th Krylov Galerkin iterate in the affine space $x_{0} + K_{m} (A, r_{0})$ for a Hermitian positive-definite system).

[L1]

The energy inner product is $⟨ u, v ⟩_{A} = ⟨ A u, v ⟩$ (The energy inner product $⟨ u, v ⟩_{A} = ⟨ A u, v ⟩$ and energy norm $∥ u ∥_{A}$ for a Hermitian positive-definite matrix).

Proof

technique · direct

1.1F1L1algebra

For every $v \in K_{m} (A, r_{0})$ , the identity $A x_{*} = b$ gives $⟨ b - A x_{m}, v ⟩ = ⟨ A (x_{*} - x_{m}), v ⟩ = ⟨ x_{*} - x_{m}, v ⟩_{A}$ by [L1]. Hence the residual is orthogonal to $K_{m} (A, r_{0})$ if and only if the error is $A$ -orthogonal to $K_{m} (A, r_{0})$ .

2.1L1step 1.1algebra∎

Suppose $x_{m}$ and $y_{m}$ both lie in $x_{0} + K_{m} (A, r_{0})$ and both satisfy the equivalent conditions from step 1.1. Then $d : = x_{m} - y_{m}$ lies in $K_{m} (A, r_{0})$ , and both errors are $A$ -orthogonal to $d$ . Subtracting the two orthogonality relations gives $0 = ⟨ x_{*} - x_{m}, d ⟩_{A} - ⟨ x_{*} - y_{m}, d ⟩_{A} = ⟨ y_{m} - x_{m}, d ⟩_{A} = - ⟨ d, d ⟩_{A} .$ Positive definiteness of the energy inner product therefore forces $d = 0$ , so $x_{m} = y_{m}$ .

Depends on

Used by

CG is the Krylov Galerkin method, and the mth iterate uniquely minimizes the A-norm of the error over x₀+Kₘ(A,r₀) Theorem

Dependency tree · two levels

6 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

Magnus R. Hestenes and Eduard Stiefel, Methods of Conjugate Gradients for Solving Linear Systems (standard reference, not scraped)

Galerkin orthogonality turns residual orthogonality into AA-orthogonality of the error, and the Galerkin iterate is unique