Polar Decomposition Calculator

The polar decomposition factors a square matrix A into a product of an orthogonal (rotation) matrix R and a symmetric positive semi-definite matrix. For a 2×2 matrix: A = R·U (right polar) and A = V·R (left polar), where R is the rotation matrix, U is the right stretch tensor, and V is the left stretch tensor.

Polar decomposition is derived directly from the Singular Value Decomposition (SVD). If A = U·Σ·Vᵀ (SVD), then the rotation matrix is R = U·Vᵀ, the right stretch is U_polar = V·Σ·Vᵀ, and the left stretch is V_polar = U·Σ·Uᵀ. This makes SVD the standard computational route for polar decomposition.

The right polar decomposition writes A = R·U, where U is the right stretch matrix and describes deformation in the material frame before rotation. The left polar decomposition writes A = V·R, where V is the left stretch matrix describing deformation in the spatial frame after rotation. Both use the same rotation matrix R.

First, compute AᵀA and find its eigenvalues and eigenvectors to form V (SVD's right singular vectors) and singular values σᵢ = √(eigenvalues). Then compute U (SVD's left singular vectors) as uᵢ = Avᵢ/σᵢ. Finally, R = U·Vᵀ, the right stretch matrix is V·Σ·Vᵀ, and the left stretch is U·Σ·Uᵀ.

For any invertible matrix A, the polar decomposition A = R·U is unique — there is exactly one orthogonal R and one symmetric positive definite U satisfying the factorization. If A is singular (det = 0), the decomposition still exists but U is only positive semi-definite and R may not be unique.

The rotation matrix R is an orthogonal matrix (Rᵀ·R = I) with determinant +1 for a proper rotation or −1 for a reflection. In continuum mechanics, it represents the rigid-body rotation component of a deformation gradient, separated from the stretching described by U or V.

In continuum mechanics, the deformation gradient F = R·U separates pure rotation from pure stretch, which is essential for strain analysis. In computer graphics and robotics, polar decomposition is used to extract rotation from transformation matrices, enabling smooth interpolation and avoiding shear artifacts.

If det(A) < 0, the matrix includes a reflection component. The resulting R will be an improper rotation (det(R) = −1), meaning it combines rotation with a reflection. The stretch matrices U and V remain symmetric positive semi-definite regardless of the sign of det(A).