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Polar decomposition splits any invertible square matrix A into A = UP, where U is orthogonal (a pure rotation or reflection, with no stretching) and P is symmetric positive semi-definite (a pure stretch, with no rotation). It's the matrix analogue of writing a complex number in polar form as a magnitude times a direction.
This calculator computes the polar decomposition of any invertible square matrix up to 6×6, using Newton's iteration, which converges quadratically to the orthogonal factor U.
How it works
Enter a square, invertible matrix. The tool repeatedly applies Newton's iteration — averaging the current estimate with the transpose of its inverse — which converges rapidly to U. Once U is found, P is recovered directly as UᵀA.
- Click Calculate to see your results.
Examples
A symmetric positive-definite example
For A = [[3,1],[1,3]] (already symmetric positive-definite), polar decomposition gives U = the identity matrix and P = A itself, since there's no rotation component to separate out.
Who should use it
- Extracting the nearest orthogonal matrix to a given (possibly noisy or drifted) transformation matrix.
- Separating rotation from scaling/shear in a computer graphics or robotics transformation.
Industry applications
- Computer graphics and robotics (orthogonalizing rotation matrices)
- Continuum mechanics (decomposing deformation into rotation and stretch)
Advantages
- Cleanly separates a transformation into its "rotation" and "stretch" components.
- Newton's iteration converges quadratically — very few iterations are needed for high precision.
Limitations
- Only applies to invertible (non-singular) square matrices.
Common mistakes to avoid
- Trying to apply this to a singular (non-invertible) matrix, for which this particular method (Newton's iteration) doesn't apply.
- Assuming U is always a proper rotation — if A has a negative determinant, U represents a reflection rather than a pure rotation.
Best practices
- Verify your result by checking that U·P reconstructs your original matrix A, and that UᵀU is (numerically) the identity matrix.
Tips
- If your matrix is already orthogonal, polar decomposition will return U equal to your original matrix and P equal to the identity matrix.