Autonne-Takagi factorization ($M=U\mathrm{diag}(s_j)U^T$)

Every complex symmetric matrix admits a unitary congruence decomposition with a unitary matrix $U$ and nonnegative diagonal entries $s_j$ equal to its singular values. This differs from unitary similarity diagonalization. It reduces a multimode squeezed vacuum to independent oscillator squeezing factors.