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2×2 QR Decomposition Calculator — result sheet
Factor a nonsingular real 2×2 matrix into an orthogonal Q matrix and an upper-triangular R matrix, then check reconstruction and orthogonality errors.
Inputs used
Results
Visual chart
Breakdown
Calculation steps
Returned data table
Formula and methodology
Formula: For columns v₁ and v₂, q₁ = v₁/‖v₁‖, r₁₁ = ‖v₁‖, r₁₂ = q₁·v₂, u₂ = v₂−r₁₂q₁, q₂ = u₂/‖u₂‖, and r₂₂ = ‖u₂‖; A = QR.
QR decomposition separates a matrix into an orthogonal factor that preserves lengths and an upper-triangular factor that carries the remaining scale and shear. This page uses a transparent two-column Gram–Schmidt contract and reports the matrices plus numerical checks, so a visitor can see whether the factorization was well-conditioned for the supplied values.
This result follows the calculator's declared inputs, precision, validation boundaries, and model limits.
Input contract
- Matrix a₁₁ — minimum -1000000; maximum 1000000
- Matrix a₁₂ — minimum -1000000; maximum 1000000
- Matrix a₂₁ — minimum -1000000; maximum 1000000
- Matrix a₂₂ — minimum -1000000; maximum 1000000
Worked example
| Input | Value |
|---|---|
| Matrix a₁₁ | 3 |
| Matrix a₁₂ | 1 |
| Matrix a₂₁ | 1 |
| Matrix a₂₂ | 2 |
Q ≈ [[0.948683,−0.316228],[0.316228,0.948683]], R ≈ [[3.162278,1.581139],[0,1.581139]], with near-zero reconstruction and orthogonality errors.
Assumptions and limits
- The input is a real 2×2 matrix with linearly independent columns.
- The calculation uses classical Gram–Schmidt with positive diagonal entries in R for this full-rank case.
- Q is orthogonal up to floating-point rounding, so QᵀQ should be close to the identity matrix.
- The page does not accept rectangular matrices, complex entries, rank-deficient inputs, or a least-squares response vector.
- Near-collinear columns are rejected when their computed second norm is below a scale-aware tolerance.
- The reconstruction check compares the entered matrix with Q×R and is a numerical diagnostic, not a proof of exact arithmetic.
- QR factorization is a matrix operation and does not by itself choose a statistical model or solve a user-specific application.
Calculator note
Source and methodology
Use the official WorldCalculate methodology policy for the source, formula, precision, and boundary standards behind this calculator.
Planning estimate, not financial, medical, legal, or professional advice. © WorldCalculate — reuse with attribution. Built and curated by Hassan ALRowaie.
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