Just as a complex number is r·e^{iθ}, a matrix splits into a rotation and a stretch. Polar decomposition is used in continuum mechanics (separating rotation from strain), computer animation and robotics.
How to use the Polar Decomposition Calculator
- Choose the matrix size – the matching grid of fields appears.
- Enter the entries row by row (a₁ b₁ c₁ is the first row, a₂ b₂ c₂ the second, and so on).
- Read the result; integer matrices give exact fractions.
Everything is calculated in your browser, so results appear instantly and nothing you type is sent to a server.
How it works
The singular value decomposition is computed first; the rotation part is W·Vᵀ and the stretch part V·Σ·Vᵀ.
Formula
A = U·P, U = W·Vᵀ, P = V·Σ·Vᵀ (from the SVD A = WΣVᵀ)
Worked example
Inputs — Matrix size: 2 × 2; a₁: 3; b₁: 1; a₂: -1; b₂: 2.
Results — Decomposition = A = U·P; U (orthogonal) = ⎡ 0.928477 0.371391 ⎤
⎣ −0.371391 0.928477 ⎦; P (symmetric positive semi-definite) = ⎡ 3.15682 0.185695 ⎤
⎣ 0.185695 2.22834 ⎦; Rotation or reflection? = U is a rotation (det U = 1).
Key terms
- Orthogonal matrix
- A matrix with QᵀQ = I.
Frequently asked questions
Is the polar decomposition unique?
P always is; U is unique when A is invertible.
What does det U tell me?
+1 means a rotation, −1 a reflection.
How is it related to the SVD?
It is built directly from the SVD factors.
What is the left polar decomposition?
A = P′U with P′ = UPUᵀ.