Polar Decomposition

Polar decomposition of a matrix is a factorization where is an orthogonal matrix and is a symmetric positive-semi definite matrix.

Geometric Intuition

The orthogonal matrix always represents rotation or reflection. And the positive-semidefinite matrix also represents (possibly none uniform) scaling.

polar_decomposition.png

Relationship with Singular Value Decomposition

Since is symmetric, we can also take a step further with a spectral decomposition : where and are rotations and is axis-aligned scaling. By that we get singular value decomposition.