Polar Decomposition Calculator — 2×2 Matrix A = QS
Enter a non-singular 2×2 matrix A to compute its polar decomposition A = QS: an orthogonal factor Q (a pure rotation or reflection) and a symmetric positive semidefinite factor S (a pure stretch). Results include both factor matrices, singular values, and the rotation angle.
Angle of the orthogonal (rotation/reflection) factor Q
How does this calculator work?
For a 2×2 matrix A, polar decomposition A = QS gives orthogonal Q (rotation/reflection) and symmetric PSD S (stretch). Compute S = (AᵀA + σ₁σ₂·I)/(σ₁+σ₂) where σ₁,σ₂ are singular values, then Q = A·S⁻¹. Enter any non-singular 2×2 matrix to get both factors instantly.
Formula
How this is calculated
The polar decomposition A = QS of a real 2×2 matrix A uniquely factorises A into an orthogonal matrix Q (det = +1 for a rotation, det = −1 for a reflection) and a symmetric positive semidefinite matrix S that captures the stretching/scaling action. It exists for any non-singular matrix.
The algorithm proceeds in four steps. First, compute M = AᵀA, which is symmetric positive semidefinite. Second, find the eigenvalues λ₁ ≥ λ₂ ≥ 0 of M; their square roots σ₁, σ₂ are the singular values of A. Third, compute S = (M + σ₁σ₂·I) / (σ₁+σ₂) — this is the exact closed-form square root of a 2×2 symmetric PSD matrix, derived from the Cayley-Hamilton theorem. Finally, Q = A·S⁻¹.
The decomposition is closely related to SVD: if A = UΣVᵀ, then Q = UVᵀ and S = VΣVᵀ. Singular values σ₁ ≥ σ₂ > 0 are the principal stretch ratios of A — how much it scales space along its two principal directions. The decomposition requires det(A) ≠ 0; singular matrices do not have a unique polar decomposition.
Frequently asked questions
It separates the rotational part of a linear map from its stretching part — essential in continuum mechanics (decomposing deformation gradient into rotation and stretch tensors), computer graphics (extracting rotation from a transformation matrix), and numerical linear algebra (computing matrix square roots).
SVD writes A = UΣVᵀ with two different orthogonal matrices and a diagonal Σ. Polar decomposition writes A = QS = (UVᵀ)(VΣVᵀ) with one orthogonal factor and one symmetric factor. SVD shows the full stretch-rotate-stretch picture; polar decomposition merges the two rotations into one.
Negative det(A) means A includes a reflection. The orthogonal factor Q will have det(Q) = −1 (an improper rotation / reflection), and the rotation angle shown is the angle of the reflective rotation. The symmetric factor S is still positive semidefinite regardless.
Also known as
TG we-Calculate Editorial Team. (2026). Polar Decomposition Calculator — 2×2 Matrix A = QS [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/polar-decomposition-calculator
TG we-Calculate Editorial Team. "Polar Decomposition Calculator — 2×2 Matrix A = QS." TG we-Calculate. 2026. https://we-calculate.com/calculator/polar-decomposition-calculator.
TG we-Calculate Editorial Team, "Polar Decomposition Calculator — 2×2 Matrix A = QS," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/polar-decomposition-calculator
@misc{wecalculate_polar_decomposition_calculator, title = {Polar Decomposition Calculator — 2×2 Matrix A = QS}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/polar-decomposition-calculator}}, year = {2026}, note = {TG we-Calculate} }
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