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.
Free theory articleFor 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.
Concept & interpretation
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
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 to2, andA22 to0. - Set the comparison vector to study:Vector x1 to
1andVector x2 to1. - 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 to0,A21 to0, and A22 to-2. Notice that det(Q) = -1, illustrating that Qincludes a reflection, not just a rotation.