Polar Decomposition

For a real nonsingular 2x2 A compute right polar A=QH, H=(AᵀA)^(1/2), Q=AH^-1. H is SPD and Q orthogonal; detA<0 gives a reflection factor, not a rotation-only story.

The polar decomposition factors a real, square, nonsingular matrixA into the product of an orthogonal matrixQ and a symmetric positive-definite matrixH:A = Q H

This is analogous to the polar form of a complex number z = r e^{iθ}. Here, H corresponds to the positive scalar r(representing a generalized "stretch" or magnitude), and Qcorresponds to the complex exponential phase factor (representing an orthogonal transformation like a rotation or reflection). When A is nonsingular, this decomposition is unique.

Worked Example

Let us calculate the right polar decomposition for the nonsingular matrixA = [[0, -3], [2, 0]].

Compute M = A^T A:

A^T = [[ 0, 2], [-3, 0]] M = A^T A = [[ 0, 2],   *  [[0, -3], [-3, 0]]       [2,  0]] = [[4, 0], [0, 9]]

Find the principal square root H = M^(1/2):
Since M is diagonal with positive entries, its principal square root is simply the diagonal matrix with the square roots of its diagonal entries:
```
H = [[√4,  0], [ 0, √9]] = [[2, 0], [0, 3]]
```

Compute the orthogonal factor Q = A H^(-1):

H^(-1) = [[1/2, 0], [0, 1/3]] Q = [[0, -3],  *  [[1/2, 0], [2,  0]]      [0, 1/3]] = [[0, -1], [1,  0]]

Verify properties:
H is symmetric positive-definite.
Q^T Q = [[0, 1], [-1, 0]] * [[0, -1], [1, 0]] = [[1, 0], [0, 1]] = I.
det(Q) = 0 - (-1) = 1, so Q is a pure rotation (by 90 degrees).
Reconstruction: Q H = [[0, -1], [1, 0]] * [[2, 0], [0, 3]] = [[0, -3], [2, 0]] = A.

Experiment

You can visualize how polar decomposition separates the transformation Ainto a stretch H followed by an orthogonal mapping Qusing the Polar Decomposition module.

Set the components of matrix A:A11 to 0,A12 to -3,A21 to 2, andA22 to 0.
Set the comparison vector to study:Vector x1 to 1 andVector x2 to 1.
Observe the "Factors" section to see the resulting Hand Q matrices, confirming our worked example.
Change the Stage to 1 Stretch H. The original vector [1, 1]is stretched along the principal axes to [2, 3].
Change the Stage to 2 Orthogonal Q. The stretched vector is then rotated by 90 degrees to become [-3, 2], which is exactly A applied to x.
Next, try setting A11 to 3, A12 to 0,A21 to 0, and A22 to -2. Notice that det(Q) = -1, illustrating that Qincludes a reflection, not just a rotation.

Limitations

There are several notable constraints in this implementation that affect how it models matrices and its numerical accuracy.

Singular Matrices: This module explicitly assumes the input matrix A is nonsingular (its determinant is non-zero). For a singular matrix, the polar decomposition still exists, but the orthogonal matrix Qis not unique. The module will display an "Unsupported singular input" status and refrain from computing a unique factorization.
Finite Analytical Calculation: The visualization uses a finite, closed-form analytical calculation for 2x2 matrices (solving for the eigenvalues of A^T A). In practice, for larger matrices, numerical iterative methods (like the Newton iteration or SVD-based methods) are required to compute the decomposition safely.
Rounding Errors: Floating-point calculations can introduce tiny rounding errors. Displays round to six decimal places; the residual is the largest absolute entry of the computed matrix A − QH. Entries and residual checks can display tiny floating residuals (for example, 1e-15) as zero; this display does not constitute an exact symbolic identity.

References

The underlying concepts of this linear algebra tool were cross-verified using materials from standard educational institutions and authors specializing in numerical methods. The references describe polar decomposition for non-singular square matrices.

Higham, N. J. (2020).What Is the Polar Decomposition?. Nicholas J. Higham's blog. This article discusses its properties and uniqueness for nonsingular matrices.

Polar Decomposition