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QR Factorization Calculator

Householder QR factorization of a small real matrix, square or rectangular. Same engine as Linear System. Not a CAS. Runs locally.

Instant result
Result
—

Enter values to calculate.

Inputs—
Mode—
Formula—
Trust summary CVP VERIFIED · 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 = QR for a small real matrix (square or rectangular, m,n ∈ [2,4]).
Scope
Real matrices
Verification
Engine tested · Specification checked · v1.14.12 · CVP VERIFIED · CVP protocol 1.0.0-proposed · Core assurance · all published modes· View Manifest · CVP overview · Specification
Versions
Calculation 1.14.12 · CVP protocol 1.0.0-proposed · Evidence 2026-10-09.qr-root-position-disks
Verification revision
2026-10-09.qr-root-position-disks · 10/10 property · digest 0214764e3fff
Legacy regression
58/58 tests · Production surface contract 5/5
Trust layers
Verification VERIFIED · Production CURRENT · Overall CVP state: VERIFIED under CVP 1.0.0-proposed
Reference
O1 model · O2 expected_values · O2 expected_values · O2 expected_values · O2 expected_values · 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 10/10 (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
431/431 golden · 8/8 CVP boundary · 68/68 invalid · 10/10 property · 1/1 metamorphic · 48/48 cross-interface · 2/2 cross-calculator · 5/5 CVP contract · Manifest
Specification basis
  • ISO 80000-2:2019 Quantities and units — Mathematics
Specification basis
Evidence
29 legacy golden · 6 legacy boundary · legacy regression suite · 431/431 oracle-backed golden · 68/68 invalid · Artifact integrity PASS
This calculator CURRENT · Public schema 1.14.12 matches · Semantic contract ✓ · Production attested · 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 = QR
OrthogonalityQᵀQ = I
iHouseholder QR. Not Gram–Schmidt as the product algorithm. Not a CAS. A ∈ R^{m×n} with m,n ∈ [2,4]. Float64. SVD is a sibling discovery URL.

How to use

1

Enter a matrix

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

2

Read Q and R

reconstruction_residual is ‖A − QR‖_F. orthogonality_residual is ‖QᵀQ − I‖_F.

Example calculations

Common configurations with formula and result.

ϟ

Upper triangular

[[1,1],[0,1]]

Q ≈ I, R ≈ A
residual ~ 0
ϟ

Overdetermined 3×2

[[1,1],[1,2],[1,3]]

A = QR
Q is 3×3, R is 3×2

QR Factorization calculator specification

Version 1.14.12 · Engine tested

Calculation status

Review policy · Evidence

Definition
A = QR with Q orthogonal and R upper triangular (or upper trapezoidal when A is rectangular). This page is a discovery surface on the linear-algebra engine (mode=qr). Least squares reuses this factorization.
What it calculates
A = QR for a small real matrix (square or rectangular, m,n ∈ [2,4]).
Inputs
  • order or rows+cols
  • a11…
Outputs
  • Q
  • R
  • reconstruction_residual
  • orthogonality_residual
  • rows
  • cols
  • condition_number
Formula
A = QR (Householder)
Assumptions
  • Real matrices
  • m,n ∈ [2,4]
  • Full Q (m×m) and R (m×n)
  • Planar ℝ² figure when 2×2; other shapes omit it
  • Float64
  • Not a CAS
Units
  • dimensionless
Boundary conditions
  • unknown mode (including jordan-form) → INVALID_MODE
  • rows or cols outside 2–4 → INVALID_INPUT
Example
[[1,1],[1,2],[1,3]] → A=QR
Validation cases

2 published on this page · 58/58 tests · Production surface contract 5/5 · View evidence

  • order=2 a11=1 a12=1 a21=0 a22=1 → reconstruction_residual≈0
  • rows=3 cols=2 a11=1 a12=1 a21=1 a22=2 a31=1 a32=3 → reconstruction_residual≈0
Specification basis
  • ISO 80000-2:2019 Quantities and units — Mathematics
Calculation version
1.14.12

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

→ 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. › → 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. ›

Frequently asked questions

Key distinctions behind the calculation.

Is this a second QR solver?

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

Is this least squares?

No. Least squares is /calc/math/least-squares on this same engine. This page only factors A = QR.

Does rectangular mean a new Capability?

No. rows and cols grow this same math.linear_algebra contract. Solve / det / inverse / LU / Cholesky stay square.