polar decomposition

Every nonzero complex number splits as z = r e^{i theta}: a nonnegative magnitude r and a pure rotation e^{i theta}. The polar decomposition is the exact matrix analogue. Any square matrix A factors as A = U P, where U is unitary (the rotation part) and P is positive semidefinite (the stretch part). The matrix plays the role of z, U plays e^{i theta}, and P plays r.

The stretch factor is concrete: P = sqrt(A^* A), the unique positive semidefinite square root, which measures how much A scales in each direction. Then U = A P^-1 when A is invertible, a genuine rotation/reflection with no stretching of its own. So the decomposition cleanly separates what A does into deform-then-rotate.

It connects directly to the SVD. If A = W Sigma V^* is the SVD, then P = V Sigma V^* and U = W V^*, so polar and SVD are two readings of the same underlying geometry. The singular values are the eigenvalues of the stretch P, and they are the magnitudes by which A stretches along its principal axes.

Why care: U is the nearest unitary matrix to A in every unitarily invariant norm, which makes the polar factor the right tool whenever you must orthogonalize a matrix that has drifted slightly off unitary, as in re-orthonormalizing a rotation in graphics, robotics, or molecular dynamics. It also underlies continuum mechanics, where P is the stretch tensor and U the rotation of a deformation.