Polar Decomposition Calculator
Factor a real 2×2 matrix into an orthogonal matrix U and a symmetric positive-semidefinite matrix P, so that A = U P.
Matrix A
Enter four real numbers. Results update as you type.
Accepted format: optional + or – sign, digits, and a decimal point. Commas, fractions, and scientific notation are rejected to prevent ambiguous input.
Live decomposition
Right polar form: A = U P
Orthogonal factor U
U represents the rotation or reflection component.
Positive-semidefinite factor P
P represents stretching along orthogonal directions.
Reconstructed product U × P
This product should reproduce A within floating-point precision.
Verification checks
Numerical diagnostics calculated from the same canonical model as the matrices and Excel workbook.
| Check | Value | Interpretation |
|---|---|---|
| Matrix rank | 2 | Both singular values are materially nonzero. |
| Reconstruction residual ‖UP – A‖F | 2.220446e-16 | Near zero confirms the factors reproduce A. |
| Orthogonality error ‖UᵀU – I‖F | 3.640555e-16 | Near zero confirms U is orthogonal. |
| Symmetry error |p₁₂ – p₂₁| | 0 | Zero confirms that P is symmetric. |
| det(U) | 1 | +1 is a rotation; – 1 includes a reflection. |
Errors are Frobenius-norm diagnostics, not additional inputs. Small nonzero values are expected because JavaScript uses IEEE 754 double-precision arithmetic.
How to use this polar decomposition calculator
What this calculator does
This tool computes the right polar decomposition of a real 2×2 matrix. It separates the linear transformation A into an orthogonal factor U and a symmetric positive-semidefinite factor P, with the exact matrix identity A = U P up to floating-point rounding. Geometrically, U captures rotation or reflection, while P captures nonnegative stretching along perpendicular principal directions. The calculator also reports singular values and numerical verification metrics. It does not solve symbolic matrices, complex-valued entries, or matrices larger than 2×2.
When to use it
Use the calculator to separate deformation from rigid motion in a planar transformation, verify a hand calculation in a linear algebra course, inspect whether a 2D map includes a reflection, or obtain the orthogonal matrix closest to a nonsingular matrix in the standard matrix-norm sense. The SciPy polar decomposition documentation gives the same right-factor convention and explains the orthogonality and positive-semidefinite properties.
How to calculate
- The calculator opens with a complete demonstration matrix A = [2, – 3], [1, 6][2, – 3], [1, 6][5, 0], [0, 45][√5, 0], [0, √45][2.236068, 0], [0, 6.708204][0.894427, – 0.447214], [0.447214, 0.894427][2, – 3], [1, 6]