Polar decomposition of an invertible real matrix

Every invertible real matrix has the polar decompositionwhere $P$ is symmetric positive definite and $R$ is orthogonal.

Every invertible complex matrix has the unique polar decompositionwhere $P$ is a positive-definite Hermitian matrix and $U$ is a unitary matrix. Writing $P=e^B$ gives $B=\frac{1}{2}\log (M{M}^{\ast })$, which is Hermitian.