Polar Decomposition Calculator

Your details

Choose a 2x2 or 3x3 real square matrix.
Entry in row 1, column 1.
Entry in row 1, column 2.
Entry in row 2, column 1.
Entry in row 2, column 2.
Singular value σ₁
4.995561

Largest singular value of A

Singular value σ₂2.245968
det(U)0.891276
det(A)10
Residual ||A - UP||3.11708818
U matrix (rows separated by |)0.601408, -0.439363 | 0.798942, 0.898310
P matrix (rows separated by |)4.017302, 1.316368 | 1.316368, 3.224226
σ₁ (largest)4.995561
σ₂2.245968
σ₃ (3x3 only)-

A = U · P: reflection/rotation composed with a symmetric stretch.

  • U is a linear map with det(U) = 0.8913. P is the symmetric stretch factor with eigenvalues equal to the singular values.
  • All singular values are positive (4.9956, 2.2460), so A is invertible and P is positive definite.
  • Condition number σ₁/σ₂ = 2.22, which is low: the matrix is well-conditioned.
  • Reconstruction residual ||A - UP|| = 3.12e+0, confirming the factorization is numerically accurate.

Next stepTo use this in continuum mechanics, U is the rotation tensor R and P is the right stretch tensor. In machine learning, polar decomposition is used to project weight matrices onto the Stiefel manifold (the set of orthogonal matrices).

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