Polar Decomposition Calculator
Decompose a matrix into its polar form A = UP (unitary/orthogonal × positive semi-definite).
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Polar Decomposition
Every matrix A can be decomposed into the product of a unitary/orthogonal matrix U and a positive semi-definite matrix P:
\[ A = UP \]
Properties:
- U is unitary (U*U = UU* = I)
- P is positive semi-definite
- P = √(A*A)
- U = AP⁻¹ if A is invertible
- P is unique, U is unique if A is invertible
Methods to Calculate:
- Singular Value Decomposition (SVD)
- Eigenvalue Decomposition
- Iterative Methods
Applications:
- Computer graphics transformations
- Quantum mechanics operators
- Signal processing
- Image processing
- Robotics and control systems
Geometric Interpretation:
- U represents rotation/reflection
- P represents stretching/compression
- Decomposition is unique up to sign
- Related to SVD decomposition