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2×2 Polar Decomposition Calculator — result sheet
Decompose an invertible real 2×2 matrix into an orthogonal factor Q and a symmetric positive-definite stretch factor H, with error checks for A = QH.
Inputs used
Results
Visual chart
Breakdown
Calculation steps
Returned data table
Formula and methodology
Formula: For S = AᵀA, H = (S + √det(S)I)/√(tr(S)+2√det(S)); Q = AH⁻¹. The real full-rank polar decomposition is A = QH, where QᵀQ = I and H is symmetric positive-definite.
The polar decomposition separates an invertible matrix's orthogonal motion/reflection from its symmetric stretch. In the 2×2 case the positive square root of AᵀA has a compact closed form, allowing the page to show H and recover Q without a hidden general-purpose symbolic system.
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 | 2 |
| Matrix a₁₂ | 1 |
| Matrix a₂₁ | 0 |
| Matrix a₂₂ | 3 |
The tool returns an orthogonal Q and symmetric positive-definite H whose product reconstructs [[2,1],[0,3]] within floating-point residuals.
Assumptions and limits
- The input is a real, invertible 2×2 matrix.
- H is the unique symmetric positive-definite right stretch factor for the full-rank real case.
- Q may have determinant +1 or −1; an orthogonal factor is not always a pure rotation.
- The square-root formula is evaluated in floating-point arithmetic and is intended for finite numerical inputs.
- Singular matrices, complex matrices, higher dimensions, and a left-polar-factor report are outside this page's contract.
- Orthogonality and reconstruction residuals are reported so rounding and conditioning remain visible.
- The result is a mathematical decomposition, not a claim about a physical material or a unique engineering interpretation.
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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