DefinitionDefinition: AI-adaptedProof: Not applicablePipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-08-30

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.

Hermitian positive-definite matrices and Cholesky factorisation A = LL* with positive diagonal

Definition

Let $n \geq 1$ and $A \in M_{n} (C)$ ; over $R$ the same definition uses transpose instead of conjugate transpose.

The matrix $A$ is Hermitian positive definite if $A = A^{*}$ and

$x^{*} A x > 0$ for every nonzero vector $x \in C^{n}$ . A Cholesky factorisation with positive diagonal of $A$ is a factorisation $A = L L^{*},$

where $L$ is lower triangular and every diagonal entry of $L$ is a positive real number.

The Hermitian terminology matches Sesquilinear and Hermitian forms over a field with an involution, using the convention linear in the first variable, and the triangular terminology matches Upper triangular, lower triangular and diagonal square matrices over a commutative ring.

Depends on

Used by

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Sources

Tobin A. Driscoll and Richard J. Braun, Fundamentals of Numerical Computation, Section 2.9 (standard reference, not scraped)
David Bindel, CS 4220: Numerical Analysis, Blocked LU and Cholesky (standard reference, not scraped)