Polar Decomposition Calculator

Decompose a 2x2 matrix A = UP where U is unitary and P is positive semi-definite.

Enter 2x2 Matrix A

Matrix A = [[a, b], [c, d]]

Results: A = U * P

Decomposition: A = U * P
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Step-by-Step Solution

A = U * P

Understanding Polar Decomposition

The polar decomposition is a fundamental factorization in linear algebra. Any square matrix A can be written as A = UP, where U is a unitary (orthogonal for real matrices) matrix and P is a positive semi-definite Hermitian matrix. This decomposition is analogous to writing a complex number in polar form z = r*e^(i*theta).

Key Components

Matrix A

The input square matrix to be decomposed.

A = U * P

Unitary Matrix U

Represents a rotation (and possibly reflection). U^T * U = I.

U = A * P^(-1)

Positive Semi-Definite P

Represents a scaling. P is symmetric with non-negative eigenvalues.

P = sqrt(A^T * A)

A^T * A

Always symmetric positive semi-definite. Its eigenvalues are the squared singular values.

A^T * A = P^2

Algorithm

  1. Compute A^T * A (the Gram matrix).
  2. Find the eigenvalues and eigenvectors of A^T * A.
  3. Compute P = sqrt(A^T * A) using the eigendecomposition.
  4. If A is invertible, compute U = A * P^(-1).

Applications

  • Computer Graphics: Separating rotation from scaling in transformations.
  • Continuum Mechanics: Decomposing deformation into rotation and stretch.
  • Signal Processing: Matrix analysis and filtering.
  • Robotics: Analyzing rigid body motions and deformations.

Relationship to SVD

The polar decomposition is closely related to the Singular Value Decomposition (SVD). If A = U_svd * Sigma * V^T is the SVD, then P = V * Sigma * V^T and U = U_svd * V^T. This connection makes polar decomposition computable via SVD.