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

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Symmetric positive-definite preconditioning preserves a Hermitian positive-definite CG problem, and the CG bound uses the transformed condition number

Statement

Let $A \in M_{n} (C)$ be Hermitian positive definite, let $x_{*}$ solve $A x = b$ , and let $M$ be Hermitian positive definite. Write $M = C C^{*}$ , and define

$\tilde{A} : = C^{- 1} A C^{- *}, \tilde{b} : = C^{- 1} b, y : = C^{*} x, y_{*} : = C^{*} x_{*} .$

Then $\tilde{A}$ is Hermitian positive definite, the transformed system

$\tilde{A} y = \tilde{b}$

is equivalent to $A x = b$ , and

$∥ y - y_{*} ∥_{\tilde{A}} = ∥ x - x_{*} ∥_{A} .$

Consequently, if

$κ : = \frac{λ_{\max} (\tilde{A})}{λ_{\min} (\tilde{A})},$

then exact-arithmetic CG on the transformed system obeys

$∥ y_{*} - y_{m} ∥_{\tilde{A}} \leq 2 {(\frac{\sqrt{κ} - 1}{\sqrt{κ} + 1})}^{m} ∥ y_{*} - y_{0} ∥_{\tilde{A}},$

which is the preconditioned CG bound.

Facts & Assumptions

Given: A Hermitian positive-definite matrix $A$ , a Hermitian positive-definite preconditioner $M$ , and the symmetric transform $\tilde{A} = C^{- 1} A C^{- *}$ .

[F1]

Symmetric preconditioning uses the variable change $y = C^{*} x$ coming from a Cholesky factorization $M = C C^{*}$ (Left preconditioning, right preconditioning, and symmetric preconditioning of a linear system).

[L1]

Equivalent preconditioned systems have the transformed residual and error maps recorded explicitly (Invertible preconditioners give equivalent linear systems, with the transformed residuals and errors written explicitly).

[L2]

A Hermitian positive-definite matrix has a Cholesky factorization $M = C C^{*}$ with invertible $C$ (A matrix admits a Cholesky factorisation with positive diagonal exactly when it is Hermitian positive definite, and that factor is unique).

[L3]

CG obeys the Chebyshev $A$ -norm bound with the spectral condition number of its Hermitian positive-definite coefficient matrix (CG obeys the Chebyshev $A$ -norm bound in terms of the spectral condition number $κ_{2} (A)$ ).

Proof

technique · direct

1.1L2algebra

By [L2], the preconditioner has a factorization $M = C C^{*}$ with invertible $C$ . For any nonzero $z$ , $z^{*} \tilde{A} z = (C^{- *} z)^{*} A (C^{- *} z) > 0,$ because $C^{- *} z \neq 0$ and $A$ is Hermitian positive definite. Also ${\tilde{A}}^{*} = (C^{- 1} A C^{- *})^{*} = C^{- 1} A^{*} C^{- *} = C^{- 1} A C^{- *} = \tilde{A} .$ So $\tilde{A}$ is Hermitian positive definite.

2.1F1L1step 1.1algebra

By [F1] and [L1], the transformed system $\tilde{A} y = \tilde{b}$ is equivalent to $A x = b$ , and its transformed error is $e_{s y m} = y - y_{*} = C^{*} (x - x_{*})$ . Therefore $∥ y - y_{*} ∥_{\tilde{A}}^{2} = (y - y_{*})^{*} \tilde{A} (y - y_{*}) = (x - x_{*})^{*} A (x - x_{*}) = ∥ x - x_{*} ∥_{A}^{2} .$ Hence the energy norms agree under the symmetric change of variables.

3.1L3step 1.1step 2.1∎

Since step 1.1 shows that $\tilde{A}$ is Hermitian positive definite, [L3] applies to the transformed system and gives $∥ y_{*} - y_{m} ∥_{\tilde{A}} \leq 2 {(\frac{\sqrt{κ} - 1}{\sqrt{κ} + 1})}^{m} ∥ y_{*} - y_{0} ∥_{\tilde{A}},$ where $κ = λ_{\max} (\tilde{A}) / λ_{\min} (\tilde{A})$ is the spectral ratio from the Statement. By step 2.1, this is exactly the preconditioned CG estimate expressed in the original variables. The bound depends on the transformed operator $\tilde{A}$ , not on $κ_{2} (A)$ by itself.

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Sources

Jonathan Richard Shewchuk, An Introduction to the Conjugate Gradient Method Without the Agonizing Pain (standard reference, not scraped)
Richard Barrett et al., Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods (standard reference, not scraped)