PropositionStatement: Literature-sourcedProof: AI-generatedverified 2026-09-26 (gpt-6-sol)

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Before convergence, every CG denominator $p_{k}^{*} A p_{k}$ is positive

Statement

Assume the conjugate-gradient recurrence of The conjugate-gradient recurrence is defined through step $k$ for a Hermitian positive-definite matrix $A$ . If $r_{k} \neq 0$ , then $p_{k} \neq 0$ and

$p_{k}^{*} A p_{k} = ∥ p_{k} ∥_{A}^{2} > 0.$

Facts & Assumptions

Given: A Hermitian positive-definite system $A x = b$ , a CG run through step $k$ , and a nonzero residual $r_{k}$ .

[F1]

CG uses the recurrence $α_{j} = \frac{⟨ r_{j}, r_{j} ⟩}{⟨ A p_{j}, p_{j} ⟩}, r_{j + 1} = r_{j} - α_{j} A p_{j}, β_{j} = \frac{⟨ r_{j + 1}, r_{j + 1} ⟩}{⟨ r_{j}, r_{j} ⟩}, p_{j + 1} = r_{j + 1} + β_{j} p_{j}$ with $p_{0} = r_{0}$ (The conjugate-gradient recurrence).

[L1]

For Hermitian positive-definite $A$ , the energy norm satisfies $∥ u ∥_{A}^{2} = ⟨ A u, u ⟩,$ and it is positive on nonzero vectors (The energy inner product $⟨ u, v ⟩_{A} = ⟨ A u, v ⟩$ and energy norm $∥ u ∥_{A}$ for a Hermitian positive-definite matrix).

Proof

technique · direct

1.1F1inductionalgebra

We first show by induction on $j \leq k$ that $⟨ p_{j}, r_{j} ⟩ = ⟨ r_{j}, r_{j} ⟩$ . For $j = 0$ this is immediate from $p_{0} = r_{0}$ in [F1]. If it holds at $j < k$ , conjugate symmetry gives $⟨ r_{j}, p_{j} ⟩ = ⟨ r_{j}, r_{j} ⟩$ , since the latter is real. The inner product is linear in its first argument, so [F1] gives $⟨ r_{j + 1}, p_{j} ⟩ = ⟨ r_{j}, p_{j} ⟩ - α_{j} ⟨ A p_{j}, p_{j} ⟩ = 0$ . Conjugate symmetry then gives $⟨ p_{j}, r_{j + 1} ⟩ = 0$ , whence $⟨ p_{j + 1}, r_{j + 1} ⟩ = ⟨ r_{j + 1}, r_{j + 1} ⟩ + β_{j} ⟨ p_{j}, r_{j + 1} ⟩ = ⟨ r_{j + 1}, r_{j + 1} ⟩ .$ Thus the identity holds for every $j \leq k$ .

2.1L1step 1.1algebra∎

At $j = k$ , step 1.1 gives $⟨ p_{k}, r_{k} ⟩ = ⟨ r_{k}, r_{k} ⟩ > 0$ because $r_{k} \neq 0$ . Hence $p_{k} \neq 0$ . By [L1], $p_{k}^{*} A p_{k} = ⟨ A p_{k}, p_{k} ⟩ = ∥ p_{k} ∥_{A}^{2} > 0,$ which is the desired positivity of the denominator.

Depends on

Used by

Cited to discharge well-definedness by The conjugate-gradient recurrence.

Dependency tree · two levels

5 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)

Before convergence, every CG denominator pk∗Apkpk∗​Apk​ is positive