ExampleConstruction: Literature-sourcedVerification: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-08-30

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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.

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A Cholesky factorisation solves a small positive-definite system efficiently

Example

The symmetric positive-definite matrix $A = (\begin{array}{cc} 4 & 2 \\ 2 & 3 \end{array})$ has Cholesky factor $L = (\begin{array}{cc} 2 & 0 \\ 1 & \sqrt{2} \end{array}),$ and for $b = (6, 5)^{T}$ the solve $A x = b$ gives $x = (1, 1)^{T}$ .

Facts & Assumptions

Given: The displayed matrix $A$ , the candidate factor $L$ , and $b = (6, 5)^{T}$ .

[L1]

Hermitian positive-definite matrices admit a unique Cholesky factorisation with positive diagonal (A matrix admits a Cholesky factorisation with positive diagonal exactly when it is Hermitian positive definite, and that factor is unique).

[L2]

Cholesky solves $A x = b$ by two triangular substitutions (Cholesky solves Hermitian positive-definite systems and has about half the factorisation cost of LU).

Verification

technique · direct

1.1

Direct multiplication gives $L L^{T} = (\begin{array}{cc} 2 & 0 \\ 1 & \sqrt{2} \end{array}) (\begin{array}{cc} 2 & 1 \\ 0 & \sqrt{2} \end{array}) = (\begin{array}{cc} 4 & 2 \\ 2 & 3 \end{array}) = A,$ so the displayed matrix is a Cholesky factor.

givenL1algebra

2.1

Solve $L y = b$ : $2 y_{1} = 6$ and $y_{1} + \sqrt{2} y_{2} = 5$ , so $y_{1} = 3$ and $y_{2} = \sqrt{2}$ . Then solve $L^{T} x = y$ : $\sqrt{2} x_{2} = \sqrt{2}$ and $2 x_{1} + x_{2} = 3$ , hence $x_{2} = 1$ and $x_{1} = 1$ .

step 1.1L2algebra

3.1

Steps 1.1-2.1 verify the example.

step 1.1step 2.1∎

Depends on

Used by

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Dependency tree · two levels

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Sources

Tobin A. Driscoll and Richard J. Braun, Fundamentals of Numerical Computation, Example 2.9.3 (standard reference, not scraped)