🌐 Polar Decomposition Calculator
Factor matrix into positive semidefinite and orthogonal parts
Currently supports 2×2 matrices only
How to Use This Calculator
Enter Square Matrix
Input your 2×2 square matrix.
Calculate
Get polar decomposition: A = QP
View Results
See P (positive semidefinite) matrix.
Formula
A = QP or A = SQ
P and S are positive semidefinite, Q is orthogonal
Definition:
Polar decomposition factors a matrix A into a positive semidefinite matrix and an orthogonal matrix.
Right Polar:
A = QP where P = √(AᵀA) and Q = AP⁻¹
Left Polar:
A = SQ where S = √(AAᵀ) and Q = S⁻¹A
About Polar Decomposition Calculator
Polar decomposition factors a matrix into a positive semidefinite matrix and an orthogonal matrix. This decomposition is analogous to the polar form of complex numbers and has important applications in continuum mechanics and computer graphics.
When to Use This Calculator
- Continuum Mechanics: Strain and stress analysis
- Graphics: Rotation and scaling of objects
- Numerical Methods: Matrix approximations
- Control Theory: System transformations
Why Use Our Calculator?
- ✅ Simplified Computation: Works for 2×2 matrices
- ✅ Positive Semidefinite: Shows P matrix
- ✅ Educational: Helps understand polar decomposition
- ✅ Free: No registration required
Key Concepts
- Positive Semidefinite: Symmetric matrix with non-negative eigenvalues
- Orthogonal: QᵀQ = I (preserves lengths and angles)
- Uniqueness: Right polar is unique if A is invertible
- Square Root: P = √(AᵀA)
Frequently Asked Questions
What is polar decomposition?
Factorization of a matrix into positive semidefinite and orthogonal parts: A = QP or A = SQ. Similar to complex numbers: z = |z|e^(iθ).
Is polar decomposition unique?
Right polar A = QP is unique if A is invertible. If not, P is unique but Q may not be.
What's the difference from SVD?
Polar decomposition gives one orthogonal matrix (rotation) and one positive semidefinite matrix. SVD gives two orthogonal matrices and a diagonal matrix.
When is polar decomposition useful?
In continuum mechanics for analyzing deformation, and in graphics for separating rotation from scaling/reflection in transformations.