Polar decomposition

Polar decomposition allows you to factor any matrix into unitary and positive semi-definite Hermitian matrices. It can be seen as breaking down a linear transformation into a rotation or reflection and scaling in ℝⁿ. Formally, it is as follows:

for any matrix A. U is a unitary matrix and P is a positive semi-definite matrix. Let's look at an example:

Using polar decomposition, this matrix can be decomposed into:

This may not seem like much, but we took a random matrix and turned it into a reflection matrix times a scaling matrix. Pretty cool!

Again, I will not go through the algorithm here because we will use calculators. Calculators for polar decomposition are not as plentiful as SVD, but I have found using the SciPy Python library to be the best way.