Polar Decomposition Calculator
Free Polar decomposition Calculator for fractions. Enter values to get step-by-step solutions with formulas and graphs.
Formula
A = UP
Where U is an orthogonal (unitary) matrix representing rotation/reflection, and P is a symmetric positive semi-definite matrix representing stretching. P = sqrt(A^T A) and U = A P^(-1).
Worked Examples
Example 1: 2x2 Matrix Polar Decomposition
Problem: Find the polar decomposition of A = [[3, 1], [2, 4]].
Solution: Step 1: Compute A^T A = [[13, 11], [11, 17]]\nStep 2: Eigenvalues of A^T A: 27.236 and 2.764\nStep 3: Singular values: sigma1 = 5.219, sigma2 = 1.663\nStep 4: P = (A^T A + sigma1*sigma2*I) / (sigma1+sigma2)\nStep 5: U = A * P^(-1)\nVerify: U is orthogonal (U^T U = I) and P is symmetric positive definite
Result: A = UP where det(U) = +1 (proper rotation), singular values: 5.219, 1.663
Example 2: Rotation Matrix (No Stretch)
Problem: Find the polar decomposition of a 45-degree rotation matrix [[0.707, -0.707], [0.707, 0.707]].
Solution: A^T A = [[1, 0], [0, 1]] = I (identity)\nBoth singular values = 1\nP = I (no stretching)\nU = A (the rotation itself)\nThis confirms that a pure rotation has P = I
Result: U = A (rotation by 45 degrees), P = I (identity, no stretch)
Frequently Asked Questions
What is polar decomposition of a matrix and what does it represent?
Polar decomposition factors a matrix A into the product of a unitary (orthogonal) matrix U and a positive semi-definite Hermitian (symmetric) matrix P, written as A = UP. This is analogous to the polar form of complex numbers, where z = r times e to the i-theta separates magnitude from rotation. Similarly, in matrix polar decomposition, U represents the rotational component (and possibly reflection) while P represents the stretching or scaling component. Every real or complex square matrix has a polar decomposition, and if A is invertible, the decomposition is unique. This factorization provides deep insight into how a linear transformation affects geometric shapes in space.
How is polar decomposition different from SVD (Singular Value Decomposition)?
Both decompositions are closely related but structured differently. SVD factors a matrix as A = U times Sigma times V-transpose, where U and V are orthogonal matrices and Sigma is a diagonal matrix of singular values. Polar decomposition factors A as A = UP, where U is orthogonal and P is symmetric positive semi-definite. The relationship between them is that P equals V times Sigma times V-transpose, and the polar U equals the SVD U times V-transpose. While SVD provides three separate factors that completely diagonalize the transformation, polar decomposition provides a more intuitive two-factor split into rotation and stretch. Both decompositions use the same singular values as fundamental building blocks.
What are the properties of the U and P factors in polar decomposition?
The U factor is an orthogonal matrix, meaning U-transpose times U equals the identity matrix and the determinant of U is plus or minus one. If the determinant of A is positive, U is a proper rotation matrix with determinant one. If the determinant is negative, U includes a reflection. The P factor is a symmetric positive semi-definite matrix, meaning P equals P-transpose and all eigenvalues of P are non-negative. If A is invertible, then P is actually positive definite with strictly positive eigenvalues. The eigenvalues of P are exactly the singular values of A, and its eigenvectors define the principal stretch directions. These properties make the decomposition geometrically meaningful and numerically useful.
How do you compute the polar decomposition of a 2x2 matrix?
For a 2x2 matrix A, first compute A-transpose times A to get a symmetric matrix. Find the eigenvalues of this product, which are the squares of the singular values sigma-1 and sigma-2. Then compute P using the formula P = (A-transpose A + sigma-1 times sigma-2 times I) divided by (sigma-1 + sigma-2). Finally, compute U = A times P-inverse. For numerical stability, you should check that sigma-1 plus sigma-2 is not too close to zero, which would indicate a nearly zero matrix. This direct formula avoids the need for a full SVD computation and works efficiently for the 2x2 case. For larger matrices, iterative methods like the Newton iteration are typically used.
What are the applications of polar decomposition in computer graphics and animation?
In computer graphics, polar decomposition is essential for extracting meaningful rotation and scaling from transformation matrices. When interpolating between two transformations (like blending animations), directly interpolating matrix entries produces unnatural artifacts like shearing. By decomposing each matrix into rotation U and stretch P, you can separately interpolate the rotations (using spherical linear interpolation) and the stretches (using linear interpolation). This produces natural-looking blends that correctly separate spinning motion from size changes. Polar decomposition is also used in physics simulations for extracting rotation from deformation gradients, and in mesh deformation algorithms where preserving local rotational behavior is critical for realistic results.
How does polar decomposition relate to the condition number of a matrix?
The condition number of a matrix, which measures its sensitivity to numerical perturbations, is directly related to the singular values that appear in the polar decomposition. The condition number equals the ratio of the largest to the smallest singular value (sigma-max divided by sigma-min). Since the eigenvalues of P in the polar decomposition are exactly the singular values of A, the condition number can be read directly from P. A condition number close to 1 means the matrix preserves relative distances (nearly orthogonal stretching), while a large condition number indicates extreme stretching in some directions relative to others. This makes the P factor a visual indicator of numerical conditioning.