Polar Decomposition Calculator

What is Polar Decomposition of a Matrix?

In linear algebra, the polar decomposition of a square matrix $A$ is a fundamental matrix factorization analogous to the polar form of a complex number $z = r e^{i\theta}$.

Just as any complex number can be expressed as the product of a non-negative magnitude $r \ge 0$ and a unit rotation $e^{i\theta}$, any real square matrix $A \in \mathbb{R}^{n \times n}$ can be factored into:

$$A = U \cdot P \quad \text{(Right Polar Decomposition)}$$

or

$$A = P' \cdot U \quad \text{(Left Polar Decomposition)}$$

where:

• $U$ is an orthogonal matrix (or unitary matrix in the complex case), meaning $U^T U = I$ and $\det(U) = \pm 1$. Geometrically, $U$ represents a pure rotation or reflection.
• $P = \sqrt{A^T A}$ is a symmetric positive semi-definite matrix representing pure stretching / scaling along orthogonal principal axes.
• $P' = \sqrt{A A^T} = U P U^T$ is the corresponding left positive semi-definite stretch factor.

How to Calculate Polar Decomposition

The most numerically stable and standard method to calculate the polar decomposition is via Singular Value Decomposition (SVD):

1. Singular Value Decomposition (SVD): Compute the SVD of $A = W \Sigma V^T$, where $W$ and $V$ are orthogonal matrices and $\Sigma = \operatorname{diag}(\sigma_1, \dots, \sigma_n)$ contains the non-negative singular values $\sigma_i \ge 0$.
2. Unitary Factor ($U$): Set $U = W V^T$.
3. Right Stretch Factor ($P$): Set $P = V \Sigma V^T = \sqrt{A^T A}$.
4. Left Stretch Factor ($P'$): Set $P' = W \Sigma W^T = \sqrt{A A^T}$.

Geometric Interpretation in Mechanics and Graphics

Polar decomposition plays a critical role in continuum mechanics, robotics, computer graphics, and elasticity theory. When a physical body undergoes deformation described by deformation gradient tensor $F$:

• $F = R U$: The deformation is decomposed into pure stretch $U$ followed by rigid-body rotation $R$.
• $F = V R$: The deformation is decomposed into rigid-body rotation $R$ followed by spatial stretch $V$.

Related Matrix and Linear Algebra Tools

Explore other matrix factorization and linear algebra calculators:

• Diagonalize Matrix Calculator: Factorize square matrices into $P D P^{-1}$.
• Cholesky Decomposition Calculator: Factor symmetric positive-definite matrices into $L L^T$.
• Characteristic Polynomial Calculator: Find matrix eigenvalues and trace.