Polar Decomposition Calculator
Decompose any square matrix A into A = UP, where U is orthogonal/unitary and P is positive semidefinite
Matrix Input
Currently supports 2×2 real matrices
Enter real numbers. Complex numbers (with i notation) have limited support.
Polar Decomposition Results
Original Matrix A:
Orthogonal Matrix U:
Positive Semidefinite Matrix P:
Verification (A = U × P):
✓ Polar decomposition successful! The matrix A has been decomposed into A = UP where U is orthogonal and P is positive semidefinite.
Example: Simple 2×2 Matrix
Try this example:
Matrix A = [[1, 2], [3, 4]]
This will decompose into orthogonal matrix U and positive semidefinite matrix P.
Properties
Orthogonal Matrix
U^T × U = I (identity matrix)
Preserves lengths and angles
Positive Semidefinite
P = P^T and all eigenvalues ≥ 0
Represents scaling and shearing
Always Exists
Every matrix has a polar decomposition
Uniqueness depends on invertibility
Tips
For invertible matrices, polar decomposition is unique
P = √(A^T A) is the positive square root
Similar to polar form of complex numbers
Used in mechanics, computer graphics, and statistics
Understanding Polar Decomposition
What is Polar Decomposition?
Polar decomposition is a matrix factorization that expresses any square matrix A as the product of an orthogonal (or unitary) matrix U and a positive semidefinite matrix P: A = UP.
Mathematical Properties
- •Existence: Every square matrix has a polar decomposition
- •Uniqueness: Unique if A is invertible
- •Orthogonal U: U^T U = I for real matrices
- •Positive P: P = P^T with non-negative eigenvalues
Calculation Method
1. Compute A^T A (or A* A for complex)
2. Find P = √(A^T A)
3. If A is invertible: U = A P^(-1)
4. Otherwise: Use SVD method
Applications
- Computer Graphics: Decompose transformations
- Mechanics: Separate rotation and deformation
- Statistics: Principal component analysis
- Signal Processing: Matrix factorization