Polar Decomposition
Decompose any 2x2 real matrix into A = UP where U is orthogonal and P is positive semidefinite. Free online polar decomposition calculator with step-by-step results for linear algebra.
About This Calculator
This polar decomposition calculator factors any invertible 2x2 real matrix A into the product A = U·P, where U is an orthogonal matrix (satisfying UTU = I) and P is a symmetric positive semidefinite matrix (satisfying P = PT and all eigenvalues >= 0). The decomposition is named by analogy with the polar form of complex numbers z = eitheta·r, where the orthogonal factor U corresponds to the rotation eitheta and P corresponds to the magnitude r.
Mathematically, the algorithm computes P as the unique positive semidefinite square root of ATA, then obtains U = A·P-1. For a 2x2 matrix with entries a11, a12, a21, a22, the calculator first forms the symmetric matrix S = ATA, computes its matrix square root using the closed-form formula for 2x2 symmetric positive semidefinite matrices, and then multiplies A by the inverse of P to recover U. If A is singular (det A = 0), the calculator reports that the orthogonal factor is not uniquely defined.
Polar decomposition is widely applied in continuum mechanics and computer graphics to separate deformation into a pure rotation component and a stretch/compression component. In robotics, it is used to extract the rotational part of a transformation matrix. This free online tool helps linear algebra students verify their textbook computations, engineers analyze deformation gradients, and researchers working with matrix factorizations.
Frequently Asked Questions
What is polar decomposition?
Polar decomposition factors any real matrix A into A = UP, where U is an orthogonal (unitary) matrix and P is a symmetric positive semidefinite matrix. It is analogous to writing a complex number z = e{itheta}·r in polar form, hence the name.
How is polar decomposition computed for a 2x2 matrix?
The polar decomposition is computed by first forming S = A^T A, then finding P as the unique positive semidefinite square root of S using the 2x2 matrix square root formula. Finally, U = A·P{-1} is obtained by multiplying A by the inverse of P.
What is the difference between polar decomposition and SVD?
The singular value decomposition (SVD) factors A = VSigmaW^T, while polar decomposition factors A = UP. The matrix P in polar decomposition equals WSigmaW^T from the SVD, and U = VW^T. Both decompositions are closely related, but polar decomposition groups the rotation and scaling differently.
Can polar decomposition be computed for singular matrices?
For singular (non-invertible) 2x2 matrices, the polar factor P is positive semidefinite but P^-1 does not exist, so the orthogonal factor U is not uniquely defined. This calculator reports when the input matrix is singular.
What are applications of polar decomposition?
Polar decomposition is used in continuum mechanics to separate deformation into rotation (U) and stretch (P), in computer graphics for 3D transformations, in control theory for optimal control problems, and in signal processing for array signal analysis.
Is polar decomposition unique?
For any real invertible matrix, the polar decomposition A = UP is unique. The matrix P is always the unique positive semidefinite square root of A^T A, and U is uniquely determined as A·P^-1. For singular matrices, P is unique but U is not.
Can this calculator handle complex matrices?
This calculator supports real-valued 2x2 matrices only. For complex matrices, the polar decomposition generalizes to A = UP where U is unitary and P is positive semidefinite Hermitian, which requires complex number arithmetic.