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Cholesky Factorization Calculator

Lower Cholesky factorization of a small real symmetric positive-definite matrix. Same engine as Linear System. Not LU. Not a CAS. Runs locally.

Instant result
Result
—

Enter values to calculate.

Inputs—
Mode—
Formula—
Trust summary CVP VERIFIED · production STALE · CVP protocol 1.0.0-proposed · Core assurance · all published modes
Input interpretation
Enter values to calculate.
Result
—
Verified scope
all published modes
Assurance
Core
Declared partition coverage
PASS · 19/19 declared partitions (solve, det, inverse, multiply, transpose, rank, eigen, lu, chol, qr, svd, lstsq, rref, condition, nullspace, columnspace, rowspace, leftnullspace, invalid-domain) · Matrix
Known limitations
  • Core CVP does not include live graph, viewport, or pointer interaction.
Model
A = LLᵀ for order 2–3 (engine up to 4), SPD only.
Scope
Real symmetric positive-definite matrices
Verification
Engine tested · Specification checked · v1.12.0 · CVP VERIFIED · production STALE · CVP protocol 1.0.0-proposed · Core assurance · all published modes· View Manifest · CVP overview · Specification
Versions
Calculation 1.12.0 · CVP protocol 1.0.0-proposed · Evidence 2026-09-22.vector-la-cvp
Verification revision
2026-09-22.vector-la-cvp · 5/5 property · digest 81ab2bb4ee85
Legacy regression
54/54 tests · Production surface contract 5/5
Trust layers
Verification VERIFIED · Production STALE · overall VERIFIED_STALE
CVP status
STALE · Capability production binding: STALE · Site report: STALE · Evidence changed after the last successful production attestation. Re-attestation required.
Reference
O1 model · O2 expected_values · O2 numerical_behavior
Interfaces
PASS · UI (SSR) / REST / MCP — ui-ssr is query-result HTML, not a live browser session. Error-path engine·REST·MCP 1/1 (status, code, calculation_version). SSR compared on URL-canonical requested calculations; empty query is idle (not an error) and JSON-typed object/array inputs are REST/MCP-only.
Supplemental domain review
Not performed
Named expert review
Not performed
CVP suite
20/20 golden · 8/8 CVP boundary · 14/14 invalid · 5/5 property · 6/6 cross-interface · 2/2 cross-calculator · 5/5 CVP contract · Manifest
Specification basis
  • ISO 80000-2:2019 Quantities and units — Mathematics
Specification basis
Evidence
27 legacy golden · 6 legacy boundary · legacy regression suite · 20/20 oracle-backed golden · 14/14 invalid · Artifact integrity PASS
STALE · Last attested schema matched 1.12.0 snapshot / local build · Semantic contract ✓ · Last attestation PASS · current evidence changed · re-attestation required · CVP STALE · attestation STALE — Production CURRENT withheld · Public/cache ✓ · Origin ✓
Semantic contract
PASS
Full verification

Manifest identity, reference classes, interfaces, suite, and production records.

Formulas

Core equations used by this calculator.

FactorizationA = LLᵀ
Diagonalℓᵢᵢ = √(aᵢᵢ − Σₖ<ᵢ ℓᵢₖ²) > 0
iLower Cholesky with positive diagonal. Not LDLᵀ. Not pivoted Cholesky. Not LU. Asymmetric A → NOT_SYMMETRIC. Nonpositive pivot → NOT_POSITIVE_DEFINITE. Order ≤ 4. Float64.

How to use

1

Enter a square SPD matrix

2×2 or 3×3. Larger composition lives at /workspace/matrix.

2

Read L

reconstruction_residual is ‖A − LLᵀ‖_F. Ill-conditioned matrices warn.

Example calculations

Common configurations with formula and result.

ϟ

[[4,2],[2,2]]

SPD 2×2

L = [[2,0],[1,1]]
residual ~ 0
ϟ

Wikipedia 3×3

[[4,12,−16],[12,37,−43],[−16,−43,98]]

L = [[2,0,0],[6,1,0],[−8,5,3]]
residual ~ 0

Cholesky Factorization calculator specification

Version 1.12.0 · Engine tested

Calculation status

Review policy · Evidence

Definition
A = LLᵀ factors a symmetric positive-definite matrix into a lower-triangular L with positive diagonal. This page is a discovery surface on the linear-algebra engine (mode=chol).
What it calculates
A = LLᵀ for order 2–3 (engine up to 4), SPD only.
Inputs
  • order
  • a11…
Outputs
  • L
  • reconstruction_residual
  • condition_number
Formula
A = LLᵀ
Assumptions
  • Real symmetric positive-definite matrices
  • Order ≤ 4
  • Planar ℝ² figure when order=2; order 3+ omits it
  • Float64
  • Not a CAS
Units
  • dimensionless
Boundary conditions
  • asymmetric A → NOT_SYMMETRIC
  • nonpositive pivot → NOT_POSITIVE_DEFINITE
  • rectangular A → INVALID_INPUT (lstsq-only)
  • unknown mode (including jordan-form) → INVALID_MODE
Example
[[4,2],[2,2]] → L=[[2,0],[1,1]]
Validation cases

2 published on this page · 54/54 tests · Production surface contract 5/5 · View evidence

  • order=2 a11=4 a12=2 a21=2 a22=2 → L[[2,0],[1,1]] reconstruction_residual≈0
  • order=2 a11=1 a12=2 a21=0 a22=1 → NOT_SYMMETRIC
Specification basis
  • ISO 80000-2:2019 Quantities and units — Mathematics
Calculation version
1.12.0

Other calculators in this family: Column Space Calculator, Condition Number Calculator, Least Squares Calculator, Left Nullspace Calculator, Linear System Calculator, LU Factorization Calculator, Matrix Determinant Calculator, Matrix Eigenvalues Calculator . Explore all Linear Algebra.

→ Column Space Calculator Column space (image) of a small real square matrix from RREF pivot columns. Same engine as Linear System. Trivial image is {0}. Left-nullspace is a sibling. Runs locally. › → Condition Number Calculator κ₁(A) = ‖A‖₁ ‖A⁻¹‖₁ for a small real square matrix. Same engine as Linear System. Singular matrices return infinite. Not a 2-norm product. Runs locally. › → Least Squares Calculator Linear least squares for a small real matrix, square or rectangular. Full column rank uses QR; otherwise SVD. Same engine as Linear System. Not Statistics regression. Runs locally. › → Left Nullspace Calculator Left-nullspace (cokernel) of a small real square matrix as ker(Aᵀ). Same engine as Linear System. Trivial left kernel is {0}. Not Jordan form. Runs locally. › → Linear System Calculator Solve small linear systems Ax = b, plus det, inverse, rank, LU, Cholesky, QR, SVD, least squares, RREF, condition number, nullspace, column space, rowspace, left-nullspace, and eigenvalues. Same engine as the matrix workspace. Not a CAS. Runs locally. › → LU Factorization Calculator Doolittle LU factorization with partial pivoting for a small real matrix. Same engine as Linear System. Not a CAS. Runs locally. › → Matrix Determinant Calculator Determinant of a small real matrix. Same engine as Linear System and the matrix workspace. Not a CAS. Runs locally. › → Matrix Eigenvalues Calculator Eigenvalues and eigenvectors of a small real matrix via the characteristic polynomial. Same engine as Linear System. Not a CAS. Runs locally. ›
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Frequently asked questions

Key distinctions behind the calculation.

Is this a second Cholesky solver?

No. It is mode=chol on math.linear_algebra, the same engine as /calc/math/linear-system and /workspace/matrix.

What if A is not SPD?

Asymmetric A returns NOT_SYMMETRIC. A symmetric matrix with a nonpositive pivot returns NOT_POSITIVE_DEFINITE. LU remains available for general square A.