Polar Decomposition is the factorization of a matrix \(A\) into the product of an orthogonal matrix \(U\) and a positive semi-definite matrix \(P\).

Formula:

\(A = UP\)

• \(U\) is an orthogonal matrix (its columns are orthogonal unit vectors, so \(U^T U = I\)).
• \(P\) is a positive semi-definite matrix (it's symmetric and its eigenvalues are non-negative).

The Process:

1. Calculate \(A^T A\).
2. Find the eigenvalues (\(\lambda_i\)) and eigenvectors (\(v_i\)) of \(A^T A\).
3. Construct \(P\) using the spectral decomposition: \(P = V D^{1/2} V^T\), where V contains the eigenvectors and D contains the eigenvalues.
4. Calculate the inverse of P, \(P^{-1}\).
5. Calculate the orthogonal matrix \(U\) using the formula \(U = AP^{-1}\).