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Polar Decomposition Calculator

Decompose any complex matrix into unitary and positive-definite matrices. Essential for linear algebra, quantum mechanics, and engineering applications.

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Use format: a+bi (e.g., 1+2i, 3-4i)
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Understanding Polar Decomposition

Polar Decomposition Formula:
For any complex matrix A ∈ ℂn×n:
A = U × P
where:
• U is a unitary matrix (U†U = I)
• P is a positive-semidefinite Hermitian matrix (P = P†, eigenvalues ≥ 0)

Quantum Mechanics

Used in quantum computing and quantum information theory for state transformations.

Robotics

Essential for rotation and scaling transformations in robotic arm movements.

Computer Graphics

Separates rotation from scaling in 3D transformations and animations.

Signal Processing

Used in MIMO systems and array processing for signal decomposition.

Algorithm Complexity

Method Time Complexity Space Complexity Best For
SVD Method O(n³) O(n²) General matrices
Eigenvalue Method O(n³) O(n²) Hermitian matrices
Iterative Method O(kn²) O(n²) Large sparse matrices

Key Properties

1
Uniqueness: The decomposition is unique if A is invertible.
2
Unitary Matrix: U preserves lengths and angles (isometry).
3
Positive-Definite: P has real, non-negative eigenvalues.
4
Relation to SVD: A = UΣV† = (UV†)(VΣV†) = UP