# Polar Decomposition Calculator

URL: https://speedycalculator.com/calculators/math/polar-decomposition/
Last updated: 2026-10-30

> The polar decomposition calculator writes a square matrix as A = U·P, with U orthogonal (a rotation or reflection) and P symmetric positive semi-definite (a stretch).

## Inputs

- Matrix size
- a₁
- b₁
- c₁
- a₂
- b₂
- c₂
- a₃
- b₃
- c₃

## Results

- Decomposition
- U (orthogonal)
- P (symmetric positive semi-definite)
- Rotation or reflection?

## Formula

```
A = U·P,  U = W·Vᵀ,  P = V·Σ·Vᵀ  (from the SVD A = WΣVᵀ)
```

## Explanation

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.

## 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.

## FAQ

**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.


## Sources

- [Wolfram MathWorld – Polar decomposition](https://mathworld.wolfram.com/PolarDecomposition.html)
- [MIT OpenCourseWare – 18.06 Linear Algebra (Gilbert Strang)](https://ocw.mit.edu/courses/18-06-linear-algebra-spring-2010/)
- [Jim Hefferon – Linear Algebra (free textbook)](https://hefferon.net/linearalgebra/)