DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (gpt-5.6-terra)audited 2026-09-01

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Left preconditioning, right preconditioning, and symmetric preconditioning of a linear system

Definition

Let $A x = b$ be a linear system.

If $P \in {GL}_{n} (F)$ is invertible, the left-preconditioned system is

$P^{- 1} A x = P^{- 1} b .$

If $P \in {GL}_{n} (F)$ is invertible and $y : = P x$ , the right-preconditioned system is

$A P^{- 1} y = b, x = P^{- 1} y .$

If $M$ is Hermitian positive definite, then by A matrix admits a Cholesky factorisation with positive diagonal exactly when it is Hermitian positive definite, and that factor is unique it has a Cholesky factor $M = C C^{*}$ with $C$ invertible. The corresponding symmetric preconditioning is

$C^{- 1} A C^{- *} y = C^{- 1} b, x = C^{- *} y .$

Depends on

Used by

Dependency tree · two levels

9 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

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