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2×2 Singular Value Decomposition Calculator — result sheet
Compute the singular values, orthonormal U and Vᵀ factors, rank, condition number, and reconstruction error of a real 2×2 matrix.
Inputs used
Results
Visual chart
Breakdown
Calculation steps
Returned data table
Formula and methodology
Formula: A = UΣVᵀ; singular values are the square roots of the eigenvalues of AᵀA, ordered σ₁ ≥ σ₂ ≥ 0.
This is a deliberately bounded 2×2 real SVD page. It returns the full square factors U and Vᵀ for the entered matrix, along with diagnostics that help a visitor check whether UΣVᵀ reconstructs the original values.
This result follows the calculator's declared inputs, precision, validation boundaries, and model limits.
Input contract
- Matrix a₁₁ — minimum -10000; maximum 10000
- Matrix a₁₂ — minimum -10000; maximum 10000
- Matrix a₂₁ — minimum -10000; maximum 10000
- Matrix a₂₂ — minimum -10000; maximum 10000
Worked example
| Input | Value |
|---|---|
| Matrix a₁₁ | 3 |
| Matrix a₁₂ | 1 |
| Matrix a₂₁ | 0 |
| Matrix a₂₂ | 2 |
The singular values are approximately 3.2566 and 1.8424; UΣVᵀ reconstructs the entered matrix to floating-point error.
Assumptions and limits
- The matrix is real and exactly 2×2; complex or larger matrices require a different input and numerical contract.
- The displayed U and Vᵀ signs may differ from another valid SVD because singular-vector signs are not unique.
- A small relative threshold is used to report numerical rank and a singular condition number for a zero second singular value.
- The result is a floating-point educational calculation, not a proof of numerical stability for every matrix scale or an engineering certification.
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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