Polar Decomposition Calculator (2×2 Matrix)

What Is Polar Decomposition?

Polar decomposition factors any real square matrix A into the product A = UP, where U is an orthogonal matrix representing pure rotation or reflection and P is a symmetric positive semi-definite matrix representing pure stretching or scaling. The decomposition is the exact matrix analogue of writing a complex number in polar form z = r·e^iθ: it separates the directional (rotational) component from the magnitude (deformation) component. For 2×2 matrices, all factors can be computed analytically with no iterative methods required.

The Core Formula

Given the 2×2 matrix A with scalar entries a (row 1, col 1), b (row 1, col 2), c (row 2, col 1), and d (row 2, col 2), the polar decomposition proceeds through three explicit steps:

1. Compute the symmetric factor P = √(A^TA). The matrix A^TA is always symmetric positive semi-definite, so its square root is well-defined. For a 2×2 matrix, A^TA has entries [[a²+c², ab+cd], [ab+cd, b²+d²]], and P is obtained via eigendecomposition of this product.
2. Compute the orthogonal factor U = A·P⁻¹. When A is invertible, U satisfies U^TU = I. If det(A) > 0, then det(U) = +1 (pure rotation); if det(A) < 0, then det(U) = −1 (improper rotation combined with reflection).
3. Extract the rotation angle θ = atan2(c − b, a + d). This direct formula recovers the net rotation angle encoded in U without explicitly inverting P, making it the numerically preferred approach for the 2×2 case.

Variable Definitions

• a — Top-left entry of matrix A (row 1, column 1)
• b — Top-right entry of matrix A (row 1, column 2)
• c — Bottom-left entry of matrix A (row 2, column 1)
• d — Bottom-right entry of matrix A (row 2, column 2)
• P — Symmetric positive semi-definite factor; its eigenvalues equal the singular values of A and encode all scaling information
• U — Orthogonal factor; det(U) = +1 indicates a proper rotation, det(U) = −1 indicates an improper rotation
• θ — Net rotation angle in radians encoded in U, computed as θ = atan2(c − b, a + d)

Worked Example

Consider A = [[3, −1], [1, 3]] with a = 3, b = −1, c = 1, d = 3.

Step 1: A^TA = [[3,1],[−1,3]]·[[3,−1],[1,3]] = [[10, 0],[0, 10]], so P = √10·I₂ — a uniform scaling by √10 ≈ 3.162.

Step 2: P⁻¹ = (1/√10)·I₂, so U = (1/√10)·[[3,−1],[1,3]]. Verification: U^TU = I and det(U) = (9+1)/10 = 1 (proper rotation).

Step 3: θ = atan2(c − b, a + d) = atan2(1 − (−1), 3 + 3) = atan2(2, 6) ≈ 0.3217 radians ≈ 18.43°. Matrix A encodes a counterclockwise rotation of approximately 18.4° followed by a uniform scale factor of √10.

Mathematical Derivation via SVD

Polar decomposition is most cleanly derived from the Singular Value Decomposition. If A = VΣW^T is the SVD (with V, W orthogonal and Σ diagonal with non-negative singular values), then U = VW^T and P = WΣW^T. The identity P = √(A^TA) follows directly because A^TA = WΣ²W^T and its positive square root is WΣW^T. This relationship is treated rigorously in Foster's Numerical Methods reference on SVD and in the Emory University SVD exercise set (Section 8.6). The singular values of A appear as eigenvalues of P, meaning both decompositions encode equivalent geometric information.

Applications Across Disciplines

• Aerospace attitude determination: NASA research applies polar decomposition to extract optimal rotation matrices from noisy vector sensor observations, as detailed in the NASA Technical Report on polar decomposition for attitude determination.
• Computer graphics and animation: 2D transformation matrices are decomposed into rotation and scale components to enable smooth interpolation and physically correct skinning.
• Continuum mechanics: The polar decomposition theorem for deformation gradients separates rigid-body rotation (U) from pure material stretch (P), a foundational result in finite-strain elasticity theory.
• Robotics and control: Projecting noisy or numerically drifted transformation matrices back onto the rotation group SO(2) requires extracting U from a computed polar decomposition.
• Symplectic Lie group integration: High-order structure-preserving integrators on SO(n) use polar decomposition to maintain orthogonality constraints, as described in NSF-funded research on symplectic Lie group methods.

Edge Cases and Numerical Notes

When det(A) = 0 (singular matrix), P is positive semi-definite rather than definite and P⁻¹ does not exist; the factor U is then non-unique. For nearly singular matrices, direct eigendecomposition of A^TA is preferred over numeric inversion. The rotation angle formula θ = atan2(c − b, a + d) is numerically robust for all non-degenerate inputs, as it relies only on the four scalar entries and avoids matrix inversion entirely.