How to Use Polar Decomposition Calculator
The Polar Decomposition Calculator factors a square matrix A into A = UP, where U is orthogonal and P is positive semi-definite. Select a 2×2 or 3×3 matrix size, fill in the entries, and the Polar Decomposition Calculator returns U, P, a verification product U·P, and the full computation trace.
- Choose matrix size – Select 2×2 or 3×3 from the dropdown.
- Enter matrix A – Type each element. Decimal and negative values are accepted.
- Read the decomposition – The Polar Decomposition Calculator shows U, P, and U·P in a readable matrix format.
- Check invertibility – The tool reports whether A is invertible, which determines uniqueness of the decomposition.
Adjust the entries to explore how rotation and stretch components change as A varies.
Formula & Theory - Polar Decomposition Calculator
The Polar Decomposition Calculator implements the right polar decomposition:
A = UP
P = √(AᵀA) (symmetric positive semi-definite)
U = A · P⁻¹ (orthogonal if A is invertible)| Symbol | Meaning |
|---|---|
| A | Input m×m real matrix |
| U | Orthogonal factor: UᵀU = I |
| P | Symmetric positive semi-definite factor |
| AᵀA | Gram matrix; its square root gives P |
Computing P = √(AᵀA):
The Polar Decomposition Calculator uses the Denman-Beavers iteration, which starts from X₀ = AᵀA and Y₀ = I, then repeats:
Xₙ₊₁ = ½(Xₙ + Yₙ⁻¹)
Yₙ₊₁ = ½(Yₙ + Xₙ⁻¹)This converges quadratically to P = √(AᵀA). Once P is known, U = A·P⁻¹.
Uniqueness: If A is invertible (non-zero determinant), the polar decomposition is unique. If A is singular, P still exists but U may not be unique.
Assumptions and Limits
The current Polar Decomposition Calculator supports 2×2 and 3×3 square matrices. The iteration runs for 30 steps; for most well-conditioned inputs this is more than enough. Near-zero pivots during matrix inversion (threshold 10⁻¹⁰) are treated as numerical singularities.
Use Cases for Polar Decomposition Calculator
The Polar Decomposition Calculator serves a range of applications in linear algebra and engineering:
- Computer graphics – Decompose affine transformation matrices into rotation and scale for animation interpolation (slerp).
- Continuum mechanics – The deformation gradient F = RU (right polar decomposition) separates rigid rotation from pure stretch.
- Matrix analysis – Study properties of U and P separately to understand spectral and geometric properties of A.
- Robotics – Recover the nearest rotation matrix to a noisy orientation estimate by extracting U from the polar decomposition.
- Numerical linear algebra – Use polar decomposition as a preprocessing step before computing the SVD.
After using the Polar Decomposition Calculator, you can verify that UᵀU ≈ I (U is orthogonal) and that P is symmetric and positive semi-definite by checking that all eigenvalues of P are non-negative.