Polar Decomposition Calculator
Compute the polar decomposition A = UP for 2ร2 and 3ร3 matrices
Results
Unitary Matrix U
Properties: U is unitary (U*U = I), where U* is the conjugate transpose.
Positive-Semidefinite Matrix P
Properties: P is positive-semidefinite (all eigenvalues โฅ 0) and Hermitian (P = P*).
Verification: A = UP
Verification: Multiplying U and P should give back the original matrix A.
Eigenvalues of P
Note: All eigenvalues of P should be non-negative for a valid polar decomposition.
About Polar Decomposition
The polar decomposition of a square matrix A is a factorization A = UP where:
- U is a unitary matrix (orthogonal if A is real)
- P is a positive-semidefinite Hermitian matrix
Applications: Computer graphics, continuum mechanics, quantum mechanics, and signal processing.
Formula: P = โ(A*A) and U = APโปยน (when A is invertible)
Quick Examples